# Seminars

The weekly Computational Analysis Seminar is attended by faculty, students, and visiting researchers working in one or more of the following areas of mathematics: constructive approximation theory, splines, wavelets, signal processing, image compression, shift-invariant spaces, constrained approximation and interpolation, computer-aided geometric design, and a few other related areas. If you need more information and/or want to be included on our mailing list, please email us at dylan.domel-white@vanderbilt.edu or cca@vanderbilt.edu.

Seminars are typically held at 4:00pm on Wednesdays in SC 1310.

## 2022 (Fall)

**Time:** Sep. 28

**Speaker**: Liudmyla Kryvonos, Vanderbilt University

**Title**: Almost optimal polynomial approximation on compact sets in $\mathbb{C}$

**Abstract**: We discuss recent results on almost optimal approximation on arcs and compact sets with non-empty interior:

– For a function $f$ that is piecewise analytic on a quasi-smooth arc $L$ and and $0<\sigma<1$ we construct a sequence of polynomials that converge at a rate $e^{-n^{\sigma}}$ at each point of analyticity of $f$ and are close to the best polynomial approximants on the whole $L$.

– For a function $u$ that is continuous on a compact set $K$ and harmonic in its interior we construct a sequence of polynomial approximants that give almost optimal rate of approximation on $K$, and in the interior points of $K$ converge faster than on the whole of $K$ (with a rate $e^{-n^{\sigma}}$, some $0<\sigma<1$).

**Time:** Nov. 16

**Speaker**: Lida Raibei, Vanderbilt University

**Title**: TBD

**Abstract**: TBD

## 2022 (Spring)

**Time:** Feb. 9

**Speaker**: Daniel Freeman, Saint Louis University

**Title**: Stable phase retrieval for infinite dimensional subspaces

**Abstract**: A frame (x_j) for a Hilbert space H allows for the stable reconstruction of any vector x in H from the frame coefficients (<x,x_j>). The goal of phase retrieval is to reconstruct x (up to a unimodular scalar) using only the absolute value of the frame coefficients. Phase retrieval using a frame for a finite dimensional Hilbert space is known to always be stable where as phase retrieval using a frame or a continuous frame for an infinite dimensional Hilbert space is always unstable. In this talk we introduce a generalization of phase retrieval to the setting of subspaces of Banach lattices and characterize when an infinite dimensional subspace of a real Banach lattice allows for stable phase retrieval. By further restricting to finite dimensional subspaces, these constructions provide new examples of continuous Parseval frames which do stable phase retrieval with stability constant independent of the dimension. The continuous frames can then be randomly sampled to give new constructions of random frames which with high probability do stable phase retrieval with stability constant independent of the dimension.

This talk will cover joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman, joint work with Dorsa Ghoreishi, and joint work with Timur Oikhberg, Ben Pineau, and Mitchell Taylor.

**Time:** Mar. 16

**Speaker**: Martin Genzel, Utrecht University

**Title**: The Separation Capacity of Random Neural Networks

**Abstract**: Neural networks with random weights appear in a variety of machine learning applications, most prominently as the initialization of many deep learning algorithms and as a computationally cheap alternative to fully learned neural networks. The goal of this talk is to enhance the theoretical understanding of random neural networks by addressing the following data separation problem: under what conditions can a random neural network make two classes (with positive distance) linearly separable? We will see that a sufficiently large two-layer ReLU-network with Gaussian weights and uniformly distributed biases can solve this problem with high probability. The number of required neurons in the two layers is explicitly linked to geometric properties of the two sets and their mutual arrangement. This instance-specific viewpoint allows us to overcome the curse of dimensionality (exponential width of the layers). The presented separation result will be also connected with related lines of work on approximation, memorization, and generalization.

This talk is based on joint work with Sjoerd Dirksen, Laurent Jacques, and Alexander Stollenwerk (arXiv:2108.00207).

**Time:** Mar. 30

**Speaker**: Dorsa Ghoreishi, Saint Louis University

**Title**: Discretizing the L_p norm and frame theory

**Abstract**: Given an N-dimensional subspace X^N of L_p([0,1]), we consider the problem of choosing M-sampling points which may be used to discretely approximate the L_p norm on the subspace. We are particularly interested in knowing when the number of sampling points M can be chosen on the order of the dimension N. For the case p=2 it is known that M may always be chosen on the order of N as long as the subspace X^N satisfies a natural L_\infty bound, and for the case p=\infty there are examples where M may not be chosen on the order of N. We show for all 1\leq p<2 that there exist classes of subspaces of L_p([0,1]) which satisfy the L_\infty bound, but where the number of sampling points M cannot be chosen on the order of N. We show as well that the problem of discretizing the L_p norm of subspaces is directly connected with frame theory.

In particular, we prove that discretizing a continuous frame to obtain a discrete frame which does stable phase retrieval requires discretizing both the L_2 norm and the L_1 norm on the range of the analysis operator of the continuous frame. This is joint work with Daniel Freeman.

**Time:** Apr. 6

**Speaker**: Ray Centner, University of South Florida

**Title**: Digital Filter Design Using OPAS

**Abstract**: In the 1980s, optimal polynomial approximants in the Hardy space H2 were suggested to be used in the design of recursive digital filters. The main reason for this was because the zeros of these OPAs lie outside of D; this allows one to create a filter that is BIBO stable. Another reason was because these OPAs can be computed efficiently; the coefficients satisfy a system of linear equations for which the associated matrix is positive definite and Toeplitz. In this talk, I will survey the process of using OPAs in H2 to design a recursive digital filter. I will then discuss BIBO stability and computations of OPAs in the context of Hp ; to inspire further research in the Banach space setting, I will discuss some open problems.

## 2021 (Fall)

**Time**: September 22, 4:10pm

**Speaker**: Yunlong Feng, University at Albany

**Title**: Robust Machine Learning beyond Maximum Likelihood

**Abstract**: Robust machine learning has been drawing more and more attention in recent years. The primary concern of robust machine learning is how to predict from observations that may be severely contaminated. In this talk, in a regression setup, we investigate a robust learning scheme induced by some error criterion that has been applied in many science and engineering fields. Despite its wide applications, theoretical studies that explain the robustness and learnability of this robust learning scheme are quite limited, which may be due to the nonconvexity of the error criterion and the presence of a scale parameter in the error criterion. To reveal how the robust learning scheme works with contaminated data, we develop a systematic theoretical framework showing that (1) the resulting regression estimator approaches the conditional mean function under weak moment conditions with a diverging scale parameter, (2) the estimator may approach the conditional mode function with a diminishing scale parameter, and (3) the robust learning scheme can learn in the presence of outliers. To further reveal the mechanisms of this robust learning scheme, we show that it can be interpreted as a minimum distance estimation approach, as opposed to the commonly used maximum likelihood estimation approach in robust machine learning. These findings, on one hand, provide a theoretical underpinning for this robust learning scheme, on the other hand, deepen our understanding of bounded nonconvex loss functions in machine learning.

**Time**: September 29, 4:00pm

**Location: **SC 1313 (in person)

**Speaker**: Minh Vu, Vanderbilt University

**Title**: Qualifying Exam – On polarization (Chebyshev) problems with external fields

**Time**: October 6, 4:10pm

**Location: **SC 1313 (in person)

**Speaker**: Rocío Díaz Martín, Vanderbilt University

**Title**: The initial state reconstruction problem: Dynamical Sampling and Observability

**Abstract**: In this talk we will start by presenting the Dynamical Sampling Problem and the Observability Problem, where the second one comes from the field Control Theory. We will establish a dictionary between them, and analyze the discrete-time and continuous-time versions. Finally, we will propose a method to estimate the initial state of a linear dynamical system with noisy observation, based on ideas from Kalman filter methods used to recover final states.

**Time**: October 13, 4:10pm

**Speaker**: Ron Levie, University of Munich

**Title**: Generalization and stability of Graph Convolutional Neural Networks

**Abstract**: Graph neural networks (GNN) are generalizations of grid-based deep learning techniques to graph structured data. The field of GNNs has grown tremendously in the past few years, leading to many practical applications with commercial impact. In this talk we focus on spectral graph convolutional neural networks, where convolution is defined as element-wise multiplication in the frequency domain of the graph, and review the mathematical foundations of their generalization capabilities. In machine learning settings where the dataset consists of signals defined on many different graphs, the trained GNN should generalize to signals on graphs outside the training set. It is thus important to transfer trained filters from one graph to the other. GNN transferability, which is a certain type of generalization capability, can be loosely defined as follows: if two graphs represent the same underlying phenomenon, then a single filter/GNN should have similar repercussions on both graphs. In this talk we will discuss the different approaches to model mathematically the notions of “graphs representing the same phenomenon” and “a filter/GNN having similar repercussions on graphs.” We will then derive corresponding transferability error bounds, proving that spectral methods are transferable.

**Time**: October 27, 4:10pm

**Location**: SC 1313 (in person)

**Speaker**: Nate Tenpas, Vanderbilt University

**Title**: Qualifying Exam – Some Optimal Periodic Configurations

**Time**: November 3, 4:10pm

**Speaker**: Alex Cloninger, University of California – San Diego

**Title**: Incorporating Invariance to Reduce the Complexity of Parametric Models

**Abstract**: Many scientific problems involve invariant structures, and learning functions that rely on a much lower dimensional set of features than the data itself. Incorporating these invariances into a parametric model can significantly reduce the model complexity, and lead to a vast reduction in the number of labeled examples required to estimate the parameters. We display this benefit in two settings. The first setting concerns ReLU networks, and the size of networks and number of points required to learn certain functions and classification regions. Here, we assume that the target function has built in invariances, namely that it only depends on the projection onto a very low dimensional, function defined manifold (with dimension possibly significantly smaller than even the intrinsic dimension of the data). We use this manifold variant of a single or multi index model to establish network complexity and ERM rates that beat even the intrinsic dimension of the data. We should note that the corollary of this result is developing intrinsic rates for a manifold plus noise data model without needing to assume the distribution of the noise decays exponentially. The second setting for building invariances concerns linearized optimal transport (LOT), and using it to build supervised classifiers on distributions. Here, we construct invariances and bound the error for deformations from various families of group actions, and show that LOT can learn a classifier on group orbits using a simple linear separator. We demonstrate the benefit of this on MNIST by constructing robust classifiers with only a small number of labeled examples. This talk covers joint work with Timo Klock and Caroline Moosmueller.

**Time**: November 10, 4:10pm

**Speaker**: Bernhard Bodmann, University of Houston

**Title**: An adaptive scattering transform for detecting anomalies in traffic counts with underlying periodicities

**Abstract**: This talk is concerned with detecting anomalies in time series using a technique from machine learning. The concrete task is to spot unusual days in single induction-loop sensor counts that measure traffic intensities throughout a year. To this end, we adapt a version of Mallat’s scattering transform to traffic count data. Based on averaged counts, a graph is constructed that models a discretized torus, in which the weekly and daily periodicities are captured. A version of a graph Laplacian gives rise to a scattering transform similar to the Diffusion Scattering Transforms on graphs. The Laplacian includes information of averaged trends and volatility in the training data. Theoretical and empirical properties of the resulting adaptive scattering transform are presented. This is joint work with Iris Emilsdottir.

**Time**: November 17, 4:10pm

**Speaker**: Yvonne Ou, University of Delaware

**Title**: Application of Herglotz-Nevanlinna functions and (generalized)-exponential analysis in the computational study of composite materials

**Abstract**: This talk comprises two parts. The first part is on the system of integral/partial differential equations that models wave propagation in poro-elastic materials such as rocks and bones. The second part is on the topic of T2-imaging of brains in the study of Alzheimer’s disease. As will be made clear in the talk, a common thread going through these two topics is the exponential analysis for composite materials. Both the theory and the computational results will be presented in the talk, which is based on the joint work with C. Bi, M. Bouhrara, K. Fishbein, W. Qian, R. Spencer, H. Woerdeman and J. Xie.

**Time**: December 1, 4:10pm

**Speaker**: Peter Hinow, University of Wisconsin – Milwaukee

**Title**: Automated Feature Extraction from Large Cardiac Electrophysiological Data Sets

**Abstract**: A new multi-electrode array-based application for the long-term recording of action potentials from electrogenic cells makes possible exciting cardiac electrophysiology studies in health and disease. With hundreds of simultaneous electrode recordings being acquired over a period of days, the main challenge becomes achieving reliable signal identification and quantification. We set out to develop an algorithm capable of automatically extracting regions of high-quality action potentials from terabyte size experimental results and to map the trains of action potentials into a low-dimensional feature space for analysis. Our automatic segmentation algorithm finds regions of acceptable action potentials in large data sets of electrophysiological readings. We use spectral methods and support vector machines to classify our readings and to extract relevant features. We are able to show that action potentials from the same cell site can be recorded over days without detrimental effects to the cell membrane. The variability between measurements 24 h apart is comparable to the natural variability of the features at a single time point. Our work contributes towards a non-invasive approach for cardiomyocyte functional maturation, as well as developmental, pathological and pharmacological studies. As the human-derived cardiac model tissue has the genetic makeup of its donor, a powerful tool for individual drug toxicity screening emerges. This is a Joint work with Stacie Kroboth, Viviana Zlochiver (Advocate Aurora Health) and John Jurkiewicz (UWM).

## 2021 (Spring)

**Time**: February 3, 4:10pm

**Speaker**: Eddy Kwessi, Trinity University

**Title**: Artificial neural networks with a signed-rank objective function

**Abstract**: Electroencephalogram (EEG) is a common tool used to measure brain activities. There is evidence in the literature that epilepsy EEG data, which are time series, display a chaotic behavior. From the embedding theory in dynamical systems and under proper conditions, such time series can be used to reconstruct the phase space of the complex system they originate from. The reconstructed phase space often produces a strange attractor which contains the information about the biological state of the patient(s). Recents studies have suggested that this strange attractors could be useful biomarkers for diseases such as epilepsy. Therefore prediction hinges on the ability to compare these attractors. To fully automate the prediction into a useful tool, machine learning algorithms should be able to distinguish between spurious and real data. One simple way to achieve this is to use an objective function in artificial neural networks with a robust objective function. In this talk, we propose to analyze artificial neural networks using a signed-rank objective function as the error function. Signed-rank objectives functions are known to be robust in the presence of outliers and possess interesting statistical properties. We prove that the variance of the gradient of the learning process is bounded as a function of the number of patterns and/or outputs, therefore preventing the gradient explosion phenomenon. Simulations show that the method is particularly efficient at reproducing chaotic behaviors from biological models such as the Logistic and Ricker models. In particular, the accuracy of the learning process is improved relatively to the least squares objective function in these cases.

**Time**: February 10, 4:10pm

**Speaker**: Taufiquar Khan, University of North Carolina at Charlotte

**Title**: Machine Learning for EIT Inverse Problem

**Abstract**: In this talk, we discuss deep learning to solve the severely ill-posed and highly nonlinear electrical impedance tomography (EIT) inverse problem. We go over the problem formulation of EIT as well as go over how a deep neural network based on transposed convolution layers can reconstruct the electrical conductivity from the simulated noisy measurements. We will also present some preliminary results how DNN may improve the recovered images compared to a conventional approach using deterministic Gauss Newton method.

**Time**: February 24, 4:10pm

**Speaker**: Zhaohua Ding, Vanderbilt University Institute of Imaging Science

**Title**: Graph theoretical analysis of the human brain and beyond

**Abstract**: Understanding structure and function relations of the human brain has been a fundamental goal of neuroscience research. Starting from segregationist and integrationist views of brain functions about 150 years ago, neuroscience research has now entered a connectionist era. Thanks to the rapid advances in modern neuroimaging technologies, it has now become routinely feasible to acquire high quality brain images in vivo with spatial and temporal resolutions that allow detailed connectopic mapping. In line with the technical development, mathematical tools for understanding topologies and modeling spatiotemporal dynamics of brain networks are also proposed. In this talk, graph theoretical approaches, which are deeply grounded in mathematics, for brain network analysis will be first summarized. Variants of graph theoretical approaches, including dynamic graphs, multilayer networks, signal processing and analysis based on graph Laplacian and information field theory will be then discussed. Finally, challenges and opportunities in brain network research are outlined.

**Time**: March 3, 4:10pm

**Speaker**: Yu-Min Chung, University of North Carolina at Greensboro

**Title**: When topology meets data: An Introduction to Topological Data Analysis and its Application to Data Sciences

**Abstract**: Topological data analysis (TDA) is a rising field at the intersection of Mathematics, Statistics, and Machine Learning. Techniques from this field have proven successful in analyzing a variety of scientific problems and datasets. The main driving force in TDA is the development of persistent homology, which studies the intrinsic shape of data. Persistent homology is a mathematical construction from the field of algebraic topology, and in practice, persistence diagrams (summaries of the persistent homology) yield topological information. However, the space of persistence diagrams is notoriously difficult to work with (for instance, the mean of persistence diagrams may not be unique). Transforming persistence diagrams into vectors or functions while preserving their critical information is one of the major research areas in TDA. In this talk, we will give a brief introduction to persistent homology, and we will demonstrate methods we propose to summarize persistence diagrams. Applications to various datasets from cell biology, medical imaging, physiology, and climatology, will be presented to illustrate the methods. This talk is designed for a general audience in mathematics. No prior knowledge in algebraic topology is required.

**Time**: March 10, 4:10pm

**Speaker**: Penghang Yin, University at Albany

**Title**: Optimization for Quantizing Neural Networks

**Abstract**: Quantized neural networks are attractive due to their efficiency in terms of run time and power consumption. However, training quantized neural networks involves minimizing a piecewise constant loss function. Such a loss function has zero gradient almost everywhere, which makes the conventional gradient-based methods inapplicable. To this end, we study a class of biased first-order oracle, termed coarse gradient, for overcoming the vanished gradient issue, and establish convergence results of the coarse gradient algorithm. In addition, we introduce a simple algorithm for training quantized neural nets, which achieves the state-of-the-art accuracies in image classification without extensive tuning of hyper-parameters.

**Time**: March 17, 4:10pm

**Speaker**: David Womble, Oak Ridge National Laboratory

**Title**: An overview of AI at Oak Ridge National Laboratory and the use of AI in accelerator operations

**Abstract**: In this talk, we will begin with an overview of AI and machine learning research and applications at Oak Ridge National laboratory. We will also discuss the use of autoencoders and LSTM autoencoders to construct a model for the linear accelerator at the Spallation Neutron Source, with the goal of detecting, predicting and preventing errant beams in the superconducting linear accelerator (SCL) subsystem.

**Time**: March 31, 4:10pm

**Speaker**: Weilin Li, New York University

**Title**: Super-resolution: theory, algorithms, and quantization

**Abstract**: This talk is concerned with the inverse problem of recovering a discrete measure on the torus given only a limited number of noisy Fourier coefficients. We connect this problem to the minimum singular value of non-harmonic Fourier matrices, derive performance guarantees for subspace methods including MUSIC and ESPRIT, and present the fundamental limits of super-resolution. We then concentrate on the problem of designing an efficient and adaptive method for quantizing these Fourier coefficients to best minimize the reconstruction error achieved by standard solvers. Time permitting, we present recent work that extends these results to a statistical setting, and discuss the theoretical and computational challenges of the multi-dimensional problem. Joint work with Albert Fannjiang, C. Sinan Gunturk, and Wenjing Liao.

**Time**: April 7, 4:10pm

**Speaker**: Gustavo Rohde, University of Virginia

**Title**: Machine learning with neural networks and other generalized projections

**Abstract**: Artificial neural networks (ANNs) have long been used as a mathematical modeling method and have recently found numerous applications in science and technology, including computer vision, signal processing, and machine learning, to name a few. Although notable function approximation results exist, theoretical explanations have yet to catch up with newer developments, particularly with regards to (deep) hierarchical learning. As a consequence, numerous doubts often accompany mathematicians and neural network practitioners alike. We will explain certain properties of neural networks from the geometric point of view, likening them to generalized Radon projections. We will describe a linearization property of neural networks, and show connections to optimal transport embeddings and transforms.

**Time**: April 14, 4:10pm

**Speaker**: Cory Hauck, Oak Ridge National Laboratory

**Title**: Discontinuous Galerkin Methods and the Diffusion Limit

**Abstract**: Discontinuous Galerkin (DG) methods were first constructed for the purpose of solving kinetic transport equations. Since then, it has been realized that DG methods perform well in scattering-dominated regimes, where the solution of the transport equation can be approximated asymptotically by the solution of a much simpler diffusion equation. For this reason, DG methods continue to be popular in applications where the diffusion limit is important. The effectiveness of DG in this limit can be traced back to the additional degrees of freedom per cell it uses (when compared to finite volume methods). However, these extra degrees of freedom come at a substantial cost, especially given the fact that memory is often the limiting factor when simulating realistic problems with a kinetic description.

In this talk, I will review some of the history of DG methods and their use in radiation transport simulations. I will then present two methods for reducing the memory of the standard DG approach while still capturing the asymptotic diffusion limit. Both methods rely on a hybrid approach to solving the transport equation.

**Time**: April 21, 4:10pm

**Speaker**: Giang Tran, University of Waterloo

**Title**: Generalization Bounds for Sparse Random Feature Expansions

**Abstract**: Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. In this work, we propose the sparse random feature expansion method, which enhances the compressive sensing approach to allow for more flexible functional relationships between inputs and a more complex feature space. We provide generalization bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature expansion method outperforms shallow networks for well-structured functions and applications to scientific machine learning tasks.

**Time**: April 28, 4:10pm

**Speaker**: Mahya Ghandehari, University of Delaware

**Title**: A non-commutative viewpoint on graph signal processing

**Abstract**: The emerging field of graph signal processing aims to develop analysis and processing techniques for data that is best represented on graphs. To this end, important notions of classical signal processing should be extended to signals on graphs. Recently, graph Fourier transform was defined as a generalization of the Fourier transform on Abelian groups, and many of its properties were investigated. However, a graph is usually the manifestation of a non-commutative structure; this can be easily seen in the case of the Cayley graph of a non-Abelian group. In this talk, we investigate a new approach to develop concepts of Fourier analysis for graphs. Our point of view is inspired by the theory of non-commutative harmonic analysis, and is founded upon the representation theory of non-Abelian groups. Taking this approach, we also study the continuity of the (now classical) graph Fourier transform over a converging sequence of graphs, and generalize a recent result of Ruiz, Chamon, and Ribeiro on graphon Fourier analysis.

## 2020 (Fall)

**Time: **November 4, 4:10pm, Zoom

**Speaker: **Shusen Pu, Vanderbilt

**Title: **Stochastic Hodgkin-Huxley Models and Noise Decomposition

**Abstract: **In this talk, we will present a natural 14-dimensional Langevin model for the Hodgkin-Huxley (HH) conductance-based neuron model in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 × 28 noise coefficient matrix. We show that

- the corresponding 14D mean-field ordinary differential equation system is consistent with the classical 4D representation of the HH system;
- the 14D representation leads to a noise coefficient matrix that can be obtained cheaply on each timestep, without requiring a matrix decomposition;
- sample trajectories of the 14D representation are pathwise equivalent to trajectories of several existing Langevin models, including one proposed by Fox and Lu in 1994;
- our 14D representation give the most accurate interspike-interval distribution, not only with respect to moments but under both the L
_{1}and L_{∞}metric-space norms; and - the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models.

We combine the stochastic shielding (SS) approximation, introduced by Schmandt and Galán in 2012, with Langevin versions of the HH model to derive an analytic decomposition of the variance of the interspike intervals (ISI), based on the mean–return-time oscillator phase. We prove in theory, and demonstrate numerically, that in the limit of small noise, the variance of the ISI decomposes linearly into a sum of contributions from each directed edge. Unlike prior analyses, our results apply to current clamp rather than voltage clamp conditions. Under current clamp, a stochastic conductance-based model is an example of a piecewise-deterministic Markov process. Our theory is exact in the limit of small channel noise. Through numerical simulations we demonstrate its applicability over a range from small to moderate noise levels. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels.

**Time: **October 21, 4:10pm, Zoom

**Speaker: **Dylan Domel-White, Vanderbilt

**Title: **Uniform error bounds for one-bit phase retrieval

**Abstract: **We consider a variant of the finite-dimensional phase retrieval problem where only one bit of information may be acquired from each magnitude measurement. Many such one-bit measurements together form a binary measurement scheme, and ideally a single fixed binary measurement scheme will admit a recovery algorithm that provides uniformly accurate phase retrieval of all possible input signals from their measurements. We examine a binary measurement scheme that asks “is the input signal closer to subspace A or subspace B?” for many random pairs of subspaces A and B and present a simple recovery algorithm to estimate an input signal from its binary measurement. Additionally, we demonstrate error bounds for our recovery algorithm which give how many pairs of subspaces we must use to achieve a desired level of uniform recovery accuracy.

## 2020 (Spring)

**Time**: February 25, 4:10pm, SC1310

**Speaker**: Albert Cohen, Sorbonne

**Title**: Optimal sampling and reconstruction on general multivariate domains

**Abstract**: Motivated by non-intrusive approaches for high-dimensional parametric PDEs, we consider the general problem of approximating an unknown arbitrary function in any dimension from the data of point samples. The approximants are picked from given or adaptively chosen finite-dimensional spaces. One principal objective is to obtain an approximation which performs as good as the best possible using a sampling budget that is linear in the dimension of the approximating space. We will show that this objective can is met by taking a random sample distributed according to a well-chosen probability measure, and reconstructing by appropriate least-squares measures. We discuss these optimal sampling strategies in the adaptive context and for general non-tensor-product multivariate domains.

**Time**: January 14, 4:10pm, SC1310

**Speaker**: Armenak Patrosyan, Oak Ridge National Laboratory

**Title**: Neural network integral representations and problems related to recovery of systems of signals

**Abstract**: Machine learning techniques have generated a lot of new opportunities in recent years for solving previously unfeasible and complex problems. Although these techniques show amazing performance in practice, a lot of their properties are not fully understood, requiring rigorous mathematical exposition. Applied harmonic analysis and optimization theory provide some of the tool-sets for such inquiry as will be addressed in this talk. Part of the presentation will focus on artificial neural networks which are computationally simple parametric functions with powerful approximation properties. A non-convex regularization method will be proposed to address the challenge of neural network overparametrization, and some theoretical aspects of the method will be investigated in the context of neural network integral representations. Certain problems related to signal reconstruction under sparsity and structural assumptions will also be discussed.

## 2019

**Time**: December 5, 4:10pm, SC1320

**Speaker**: Karl Liechty, DePaul University

**Title**: Error bounds in Fourier extension approximations

**Abstract**: A truncated Fourier series is a very effective way to approximate smooth periodic functions, but if a function defined on an interval is not periodic, its nearest truncated Fourier series is not a good approximation near the endpoints of the interval due to the Gibbs phenomenon. One method to deal with this issue, known as Fourier extension or Fourier continuation, is to extend the function smoothly to one which is periodic on a slightly larger interval, and approximate by truncated Fourier series with a slightly larger period. I will discuss asymptotic error bounds for these truncated Fourier series as the number of Fourier modes becomes large. In particular I will discuss the issue of obtaining uniform bounds from a discrete L^2 construction. This is joint work with Jeff Geronimo.

**Time**: April 16, 4:10pm, SC1312

**Speaker**: David Zhang, Vanderbilt University

**Title**: Deriving New Runge-Kutta Methods Using Unstructured Numerical Search

**Abstract**: Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces, and its number of stages, which measures the amount of computational work it requires. The primary goal in constructing Runge-Kutta methods is to maximize order using a minimum number of stages. However, high-order Runge-Kutta methods are difficult to construct because their parameters must satisfy an exponentially large system of polynomial equations. In this talk, I will present the first known 10th-order Runge-Kutta method with only 16 stages, breaking a 40-year standing record for the number of stages required to achieve 10th-order accuracy. I will also discuss the tools and techniques that enabled the discovery of this method using a straightforward numerical search.

**Time**: March 20, 3:10 pm, SC1310

**Speaker**: Sergiy Borodachov, Towson University

**Title**: Optimal global recovery of three times differentiable functions

**Abstract**: We will discuss the problem of optimal global recovery of the class of three times differentiable functions with uniformly bounded third order derivatives in any direction on a d-dimensional convex polytope inscribed in a sphere and containing its circumcenter. The data known about each function are its values and gradients at the vertices of the domain. We consider the class of non-adaptive recovery algorithms and measure the recovery error in the uniform norm. The optimality of a certain quasi-interpolating recovery method is established, which becomes a quadratic polynomial when the domain is a simplex. If time permits, we will talk about a similar problem with the data of smoothness two.

**Time**: March 18, 3:10 pm CHANGED TO: 4:10pm, SC1310

**Speaker**: David de Laat, Emory University

**Title**: Pair correlation estimates for the zeros of the zeta function via semidefinite programming

**Abstract**: In this talk I will explain how sum-of-squares characterizations and semidefinite programming can be used to obtain improved bounds for quantities related to zeros of the Riemann zeta function. This is based on Montgomery’s pair correlation approach. I will show how this connects to the sphere packing problem, and speculate about future improvements. Joint work with Andrés Chirre and Felipe Gonçalves.

**Time**: February 13, 3:10 pm, SC1308

**Speaker**: Rong Liu, Fujian Agriculture & Forestry University

**Title**: Orthogonal polynomials for Jacobi-exponential weights

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## 2018

**Time**: December 5, 4:10 pm, SC 1310

**Speaker**: Johann Brauchart, Technische Universität Graz

**Title**: Minimal logarithmic energy circles on sets of revolution

**Abstract**: We discuss the discrete and continuous minimal logarithmic energy problem on compact sets of revolution. We present theoretical and numerical results for finite cylinders (revolving line segments) and circular tori (revolving circles). Even these simplest cases reveal a whole host of unresolved fundamental questions about how to characterize minimal energy configurations, the asymptotics of their potential energy, their limit distribution and its support. Taking advantage of rotational symmetry, we reduce the minimum energy problem in 3-space for the singular logarithmic kernel to a plane problem for some continuous kernel. The points solving the discrete problem for the new kernel turn into the circles in the title. This is joint work with Doug Hardin and Edward B. Saff.

**Time**: October 17, 4:10 pm, SC 1310 CHANGED TO: October 24, 4:10 pm, SC 1310

**Speaker**: Tetiana Stepaniuk, Technische Universität Graz

**Title**: Approximation by Fourier sums and hyperuniform point sets

**Abstract**: In the first part of talk we find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of generalized Poisson integrals. This problem leads to the problem of finding the asymptotic equality of Lp-norm of the remainder of trigonometric series. In the second part of talk we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J. Brauchart, P. Grabner, W. Kusner and J. Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls or balls of threshold order on the flat tori are uniformly distributed.

**Time**: October 10, 4:10 pm, SC 1310

**Speaker**: Ian Wagner, Emory University

**Title**: Thoughts on sphere packing

**Abstract**: Work of Cohn and Elkies in 2003 offered convincing evidence that the sphere packing problem was within reach for dimensions 8 and 24. In 2016 Viazovska explicitly constructed special Schwartz functions using modular forms to resolve the problem in 8 dimensions. Her method was quickly adapted to also resolve the problem in 24 dimensions. I will review Viazovska’s breakthrough and the role modular forms play in it. In particular I will discuss how to construct Schwartz functions in general using modular forms. I will end with some potential applications for these Schwartz functions to problems related to sphere packing.

**Time**: September 19, 4:10 pm, SC 1310

**Speaker**: Damir Ferizović, Technische Universität Graz

**Title**: An Upper Bound for the Green Energy on SO(3)

**Abstract**: In this short talk, we compute a simple expression for Green’s function on the Lie group SO(3) using Wigner D-functions, the eigenfunctions of the corresponding Laplace-Beltrami Operator, and use a result of Macchi-Soshnikov to calculate an upper bound for the Green energy there on. To this end, we give a small summary of Determinantal Point Processes, and if time permits, we will prove asymptotics of the L^{2}-norm of Gegenbauer (hyperspherical) polynomials. This is a joint work with Carlos Beltrán from Universidad de Cantabria.

**Time**: February 28, 3:10 pm, SC 1432

**Speaker**: John Paul Ward, North Carolina A&T State University

**Title**: Localized Basis Functions on Graphs and Applications

**Abstract**: Graph domains are being used for many signal processing applications. They provide a more general framework than integer lattices, and they can be used to incorporate additional structural or geometric information. In this talk, we define intrinsic basis functions derived from the graph Laplacian, in analogy with polyharmonic splines on euclidean spaces. We consider the associated Lagrange basis functions and discuss their decay properties. The applications of such bases include kernel-based machine learning algorithms where data is well represented using a graph framework, and we shall present some preliminary experiments in this direction.

## 2017

**Time**:April 12, 4:00 pm, SC 1432

**Speaker**: Sina Sadeghi Baghsorkhi, University of Michigan

**Title**: A new framework for numerical analysis of nonlinear systems: the significance of the Stahl’s theory and analytic continuation via Pade approximants

**Abstract**: An appropriate embedding of polynomial systems of equations into the extended complex plane renders the variables as functions of a single complex variable. The relatively recent developments in the theory of approximation of multi-valued functions in the extended complex plane give rise to a new framework for numerical analysis of these systems that has certain unique features and important industrial applications. In electricity networks the states of the underlying nonlinear AC circuits can be expressed as multi-valued algebraic functions of a single complex variable. The accurate and reliable determination of these states is imperative for control and thus for efficient and stable operation of the electricity networks. The Pade approximation is a powerful tool to solve and analyze this class of problems. This is especially important since conventional numerical methods such as Newton’s method that are prevalent in industry may converge to non-physical solutions or fail to converge at all. The underlying concepts of this new framework, namely the algebraic curves, quadratic differentials and the Stahl’s theory are presented along with a critical application of Pade approximants and their zero-pole distribution in the voltage collapse study of the electricity networks. Joint work with Nikolay Ikonomov (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences) and Sergey Suetin (Steklov Mathematical Institute, Russian Academy of Sciences).

**Time**: December 6, 3:10 pm, SC 1432

**Speaker**: Eddy Kwessi, Trinity University

**Title**: Recent Advances in Rank Estimation Methods

**Abstract**: In this talk, I will discuss rank estimation methods. I will especially make the case of why it is a method worth considering if we are concerned with robustness of our statistical estimators. I will discuss some recent applications of this method, including semi-parametric estimation with splines/or wavelets, single-index estimation, nonlinear estimation, Bayesian estimation, penalized regression. I will also show how the method of ranks can be coupled with the Amplitude Message Passing (AMP) algorithm — often use in stochastic differential equations — to solve LASSO-type problems. In this era of “big data”, this is especially important if one is interested in high dimension estimation where there are often more covariates than data points.

**Time**: November 8, 4:10 pm, SC 1432

**Speaker**: John Murray-Bruce, Boston University

**Title**: Sensing the real world: a sampling-theoretic look at regularized PDE-constrained inverse problems

**Abstract**: Partial differential equations are central to describing many physical phenomena such as: light transport, spreading of diseases, dispersion of plumes etc. In many of these applications, the phenomena are observed through a sensor network, such as an array of photosensitive detectors, with the aim of inferring certain underlying properties of the phenomena. It is well known that these classes of inverse problems can be ill-posed, thus requiring proper regularization. In the first half of this talk, we show how a class of PDE driven inverse source problems can be regularized parametrically, by exploiting source sparsity. Specifically, we explore a promising framework that leverages results from sampling and approximation theory to solve the class of inverse source problems, governed by linear partial differential equations. As we will see, the unknown field sources can be recovered from a sequence of, so called, generalized measurements by using multidimensional frequency estimation techniques. Explicit formulas to map the sensor samples into the sequence of generalized measurements when the Green’s function (approximately) satisfies the generalized Strang-Fix condition are derived. Based on this new framework, we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and present numerical simulation results to verify that they compare favorably against compressed sensing-based methods. The second half of this talk turns to the non-line-of-sight (NLOS) imaging problem, with applications scenarios with limited visibility: from automotive safety and endoscopic imaging to search-and-rescue operations. In this problem, the aim is to reconstruct a scene of interest from indirect measurements. Given that, the underlying phenomena is the transport of photons to the detector from a hidden, around-the-corner scene, we will see that this imaging problem is severely ill-conditioned, but can be conveniently regularized by exploiting the use of an occluder. We explore a sampling and approximation theoretic approach to study the field-of-view and resolution limits of the resulting computational camera. Finally to conclude the talk, numerical reconstruction results using real NLOS imaging data are presented.

**Time**: October 25, 3:10 pm, SC 1432

**Speaker**: Vignon Oussa, Bridgewater State University

**Title**: Frames arising from solvable actions

**Abstract**: In this presentation, we will provide a unified method which is exploited to construct reproducing systems arising from unitary irreducible representations of solvable Lie groups. We will show how a careful study of differential geometric properties of coadjoint orbits leads to a systematic and explicit construction to discrete frames and smooth frames of compact supports. In contrast to well-known techniques such as the coorbit theory and other discretization schemes, we do not assume the integrability or square-integrability of the representations of interest. Additionally, we will present various examples which illustrate that our method handles a variety of groups relevant to wavelet theory and time-frequency analysis.

**Time**: October 18, 4:10 pm, SC 1432

**Speaker**: Kathy Driver, University of Cape Town

**Title**: Zeros of Ultraspherical and pseudo-Ultraspherical polynomials

**Abstract**: The pseudo-ultraspherical polynomial of degree $n$ is defined by $\tilde{C}_n^{(\lambda)}(x) =(-i)^n C_n^{(\lambda)}(ix)$ where $C_n^{(\lambda)}(x)$ is the ultraspherical polynomial. We discuss the orthogonality of finite sequences of pseudo-ultraspherical polynomials $\{\tilde{C}_{n}^{(\lambda)}\}_{n=0}^{N}$ for different values of $N$ that depend on $\lambda.$ We discuss applications of Wendroff’s Theorem and use an identity linking the zeros of the pseudo-ultraspherical polynomial $\tilde{C}_n^{(\lambda)}$ with the zeros of the ultraspherical polynomial $C_n^{(\lambda’)}$ where $\lambda’= \frac12 – \lambda -n$ to prove that when $ 1-n < \lambda < 2-n,$ two (symmetric) zeros of $\tilde{C}_{n}^{(\lambda)}$ lie on the imaginary axis.

**Time**: October 4, 4:10 pm, SC 1432

**Speaker**: Kiryung Lee, Georgia Tech

**Title**: Generalized notions of sparsity and restricted isometry property: A unified framework and applications

**Abstract**: The restricted isometry property (RIP) has been an integral tool in the analysis of various inverse problems with sparsity models in signal processing and statistics. We propose generalized notions of sparsity and provide a unified framework for the RIP of structured random measurements given by isotropic group actions. Our results extend the RIP for partial Fourier measurements by Rudelson and Vershynin to a much broader context and identify a sufficient number of group structured measurements for the RIP on generalized sparsity models. We illustrate the main results on an infinite dimensional example, where the sparsity represented by a smoothness condition approximates the total variation. We also discuss fast dimensionality reduction on generalized sparsity models. In generalizing sparsity models, the parameter accounting for the level of sparsity becomes no longer sub-additive. Therefore, the RIP does not preserve distances between sparse vectors. We show a weaker version with additive distortion, which is similar to analogous property arising in the 1-bit compressed sensing problem. This is a joint work with Marius Junge.

**Time**: September 27, 4:10 pm, SC 1432

**Speaker**: Abdul Jerri, Clarkson University

**Title**: Multivariate and Some Other Extensions of Sampling Theory for Signal Processing-A Tutorial Review

**Abstract**: The well known Shannon Sampling Theorem is well known for the one dimensional signals. However, Shannon in his Seminal paper 1948 -1949 had the Multidimensional case on his mind, where in his first step of the source for his four steps communication system, he mentioned a number of multidimensional cases. To date, we have many papers, and sections or chapters in sampling books, on multidimensional sampling, and a good number of tutorials on the one dimensional case, but a tutorial review of the multidimensional case is lacking. This talk gives a summary of a detailed review paper on this valued subject, which is close to completion, The paper is dedicated for the celebration of the Centennial of the father of Communication theory, Claude Elwood Shannon.

**Time**: September 13, 4:10 pm, SC 1432

**Speaker**: Gustavo Rohde, University of Virginia

**Title**: A signal transformation approach for transport-based pattern theory

**Abstract**: We describe a new class of signal processing & data analysis techniques based on the mathematics of optimal mass transport. These techniques can be interpreted as nonlinear transforms with well defined-forward (analysis) and inverse (synthesis) operations with demonstrable advantages over standard linear transforms (Fourier, Wavelet, Radon, Ridgelet, etc.). In particular, they are able to provide parsimonious signal and image representation models which can be shown theoretically and experimentally to enhance linear separability of signal classes. We demonstrate these techniques on applications related to inverse problems and biomedical image analysis.

**Time**: March 29, 4:00 pm, SC 1432

**Speaker**: Karamatou A. Yacoubou Djima, Department of Mathematics and Statistics, Amherst College

**Title**: Diffusion Frames on Graphs

**Abstract**: Multiscale (or multiresolution) analysis is used to represent signals or functions at increasingly high resolution. In this talk, we construct frame multiresolution analyses (MRA) for $L^2$-functions of spaces of homogeneous type. In this instance, dilations are represented by operators that come from the discretization of a compact symmetric diffusion semigroup. The eigenvectors shared by elements of the compact symmetric diffusion semigroup can be used to define an orthonormal MRA for $L^2$. We introduce several frame systems that generate an equivalent MRA, notably composite diffusion frames, which are built with the composition of two “similar” compact symmetric diffusion semigroups.

**Time**: February 10, 4:00 pm, SC 1432

**Speaker**: Oussa Vignon, Bridgewater State University

**Title**: Discrete frames arising from irreducible solvable actions

**Abstract**: In this presentation, we will provide a unified method which is exploited to construct reproducing systems arising from unitary irreducible representations of solvable Lie groups. A careful study of differential geometric properties of coadjoint orbits leads to a systematic and explicit construction to discrete frames and smooth frames of compact supports. In contrast to well-known techniques such as the coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Additionally, we will present various examples which illustrate that our method handles a variety of groups relevant to wavelet and time-frequency analysis experts. For example, the ax+b group, the generalized Heisenberg groups, the Shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are just a few examples of groups that can be handled in a unified fashion by our method.

**Time**: January 20, 4:00 pm, SC 1432

**Speaker**: Galyna Livshyts, Georgia Tech

**Title**: Bounding Marginals of Product Measures

**Abstract**: It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This bound yields a sharp small ball inequality for the length of a projection of a random vector. This talk is based on the joint work with G. Paouris and P. Pivovarov.

**Time**: January 18, 4:00 pm, SC 1432

**Speaker**: Irene Waldspriger, MIT

**Title**: Phase Retrieval for the Cauchy Wavelet Transform.

**Abstract**: We consider the following problem: to what extent is it possible to reconstruct a function from its wavelet transform modulus? This question has important theoretical as well as practical motivations, coming from audio processing. It belongs to the family of inverse problems known as “phase retrieval problems”. In a first part, we will describe our theoretical results on this question: any L2 function is uniquely determined by the modulus of its wavelet transform. The reconstruction operator is continuous; it is not uniformly continuous, but satisfies a “local stability” property, that is stronger than continuity only. We will complement these results with the description of a reconstruction algorithm, and reconstruction examples. In a second part, we will see which implications these results have for the understanding of a more sophisticated, deep representation: the scattering transform, introduced by Mallat.

## 2016

**Time**: November 16, 2016, 4:10 pm, SC 1432

**Speaker**: Luc Vinet, University of Montreal

**Title**: Quantum state transport, entanglement generation and orthogonal polynomials

**Abstract**: TBA **Time**: October 19, 2016, 4:10 pm, SC 1432

**Speaker**: John Jasper, University of Cincinatti

**Title**: Equiangular tight frames from association schemes

**Abstract**: An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound. Though they arise in many applications, there are only a few known methods for constructing ETFs. One of the most popular classes of ETFs, called harmonic ETFs, is constructed using the structure of finite abelian groups. In this talk we will discuss a broad generalization of harmonic ETFs. This generalization allows us to construct ETFs using many different structures in the place of abelian groups, including nonabelian groups, Gelfand pairs of finite groups, and more. We apply this theory to construct an infinite family of ETFs using the group schemes associated with certain Suzuki 2-groups. Notably, this is the first known infinite family of equiangular lines arising from nonabelian groups.

**Time**: November 30, 2016, 4:10 pm, SC 1432

**Speaker**: Enrico Au-Yeung, DePaul University

**Title**: Tensor Networks and the Blessing of Dimensionality

**Abstract**: Have you ever looked at your thumb and admire how smart is your thumb? The protein molecules in your body can perform computation hundreds of times faster than a cluster of computers. There are three short stories that I want to tell. Tensor networks are tools that can be used to solve a wide class of data intensive problems in machine learning, physics, and signals processing. The basic idea is to turn a long vector or a large matrix into a tensor, then draw some cute diagrams. Each such diagram actually represents a formidable equation. Another story here is Optimization beyond Grandma’s Lagrange Multiplier. The term Compressed Sensing means recovering a long vector by making a small number of measurements. Until a few years ago, to do compressed sensing, you need a matrix to satisfy RIP (restricted isometry property). What if your matrix does not satisfy RIP, but you have a good toolbox for solving optimization problems? For the third story, you will have to hear it at the talk. Most of this talk will be accessible to graduate students in mathematics.

**Time**: September 16, 2016, 3:10 pm, SC 1310

**Speaker**: David Benko, University of South Alabama

**Title**: Estimating the probability of heads of a fake coin.

**Abstract**: We tossed a biased coin 10 times and we got 3 heads. What is the probability of heads? The maximum-likelihood method claims it is 0.3 but we are unhappy with that method. Using game theory we answer the question explicitly for 1 and 2 tosses.

**Time**: April 6, 2016, 3:10 pm, SC 1432

**Speaker**: Alexander Volberg, Michigan State University

**Title**: Monge–Ampère equations with drift and end-point estimates in harmonic analysis.

**Abstract**: We will review a couple of end-point estimates in harmonic analysis that can be sort of equivalently reduced to understanding of the behavior of solutions of certain MA equations with drift, where the behavior of the drift seems to be curial. As a result, some new end-point estimates for singular integrals will be proved.

**Time**: March 16, 2016, 3:10 pm, SC 1432

**Speaker**: Oleksandra Beznosova, University of Alabama

**Title**: On the star discrepancy conjecture.

**Abstract**: The L_{∞}-star discrepancy measures how well a discrete measure supported on a given set of N points approximates a uniform measure on the multidimensional unit cube, the smaller the discrepancy the better the approximation. Therefore, we are interested in the sharp lower bound on discrepancy as a function of N and optimal sets on which it is achieved. Bounds on the discrepancy are used, for example, in the error bounds for quasi-Monte Carlo methods. It is somewhat intuitive that a discrete measure cannot approximate continuous measure too well. In dimension 2 we know (up to a numerical constant) lower bounds on the discrepancy as function of N, and some examples of sets on which lower bounds are achieved. In higher dimension d>2, it is an open conjecture that optimal L_{∞}-star discrepancy is of the order N^{-1}(log N)^{d/2 }

## 2015

**Time**: November 11, 2015, 3:10 pm, SC 1432

**Speaker**: Laura De Carli, Florida International University

**Title**: Constructing new bases from old.

**Abstract**: available here

**Time**: November 11, 2015, 4:10 pm, SC 1432

**Speaker**: Grady Wright, Boise State University

**Title**: Computing with functions on the sphere using low rank approximations

**Abstract**: A collection of algorithms for computing with functions on the surface of the unit two-sphere is presented. Central to these algorithms is a new scheme for approximating functions to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. The scheme gives a low rank representation of the approximants that reduces oversampling issues near the poles, converges for certain analytic functions, and allows for stable differentiation. The low rank representation also makes operations such as function evaluation, differentiation, and integration particularly efficient. A demonstration of the algorithms, which are implemented in Chebfun, will be given.

**Time**: November 4, 2015, 3:10 pm, SC 1432

**Speaker**: Yujian Su, Vanderbilt University

**Title**: Dissertation defense

**Abstract**: TBA

**Time**: October 7, 2015, 3:10 pm, SC 1432

**Speaker**: Tim Michaels, Vanderbilt University

**Title**: Point sets on the sphere and their Riesz energies

**Abstract**: Generating suitable point sets and meshes on the sphere is a problem spanning many areas in numerical analysis. We present a survey of quickly generated point sets on S^2 which have been created for a variety of purposes, examine their equidistribution properties, separation, covering, and mesh ratio constants and derive a new point set, equal area icosahedral points, with low mesh ratio. We analyze numerically the leading order asymptotics for the Riesz and logarithmic potential energy for these configurations with total points up to 10,000.

**Time**: September 30, 2015, 3:10 pm, SC 1432

**Speaker**: Bubacarr Bah, University of Texas at Austin

**Title**: Structured sparse recovery with sparse sampling matrices

**Abstract**: Compressed sensing seeks to exploit the simplicity (sparsity) of a signal to under sample the signal significantly. Sparsity is a first order prior information on the signal. In many applications signals exhibit an additional structure beyond sparsity. Exploiting this second order prior information about the signal not only enables further sub-sampling but also improves accuracy of reconstruction. On the other hand, a lot of the sampling matrices, for which we are able to prove optimal recovery guarantees, are dense and hence do not scale well with the dimension of the signal. Sparse matrices scale better than their dense counterparts but they are more difficult to give provable guarantees on. The sparse sampling operators we consider are adjacency matrices of lossless expander graphs. They are non-mean zero and they reflect more some of the applications of compressed sensing like the single pixel camera. We also propose two reconstruction algorithms. A non-convex algorithm that converges linearly with the signal dimension and a convex algorithm that is comparable and sometimes outperforms existing popular algorithms. We also derived sharp sample complexity bounds. This talk will give a general overview of results on structured sparsity in compressed sensing (model-based compressed sensing). It will discuss sampling and recovery in model-based compressed sensing generally but will narrow down to give latest results our work on model-based compressed sensing with sparse sensing matrices from expanders.

**Time**: September 9, 2015, 3:10 pm, SC 1432

**Speaker**: Keaton Hamm, Vanderbilt University

**Title**: Sampling and Interpolation with Radial Basis Functions

**Abstract**: For some time, there have been connections between interpolation schemes involving radial basis functions and classical sampling theory. This talk will explore some of these connections both in the uniform and nonuniform settings. In the former case, the technique of cardinal interpolation seeks to approximate a given smooth function by integer translates of a single function, for example, the Gaussian kernel or Hardy multiquadric. This is similar to the classical sampling theorem which provides exact recovery of a bandlimited function via translates of sinc. However, in the nonuniform case, the problem becomes somewhat more functional analytic, and so far there are some restrictions on what type of point sets one may use in the interpolation schemes; in particular, so-called complete interpolating sequences for Paley-Wiener spaces are such an admissible set. Time permitting, we may also discuss some ways in which one may obtain approximation rates for the schemes discussed before.

**Time**: April 29, 2015, 3:10 pm, SC 1432

**Speaker**: Liao Wenjing, Duke University

**Title**: Gridding error and super-resolution in spectral estimation

**Abstract**: The problem of spectral estimation, namely – recovering the frequency contents of a signal – arises in various applications, including array imaging and remote sensing. In these fields, the spectrum of natural signals is composed of a few spikes on the continuum of a bounded domain. After the emergence of compressive sensing, spectral estimation has been widely studied with an emphasis on sparse measurements. However, with few exceptions, the spectrum considered in the compressive sensing community is assumed to be located on a DFT grid, which results in a significant gridding error.

In this talk, I will present the MUltiple SIgnal Classification (MUSIC) algorithm and some modified greedy algorithms, and show how the problem of gridding error can be resolved by these methods. Our work focuses on a stability analysis as well as numerical studies on the performance of these algorithms. Moreover, MUSIC features its super-resolution effect, i.e., the capability of resolving closely spaced frequencies. We will provide numerical experiments and theoretical justifications to show that the noise tolerance of MUSIC follows a power law with respect to the minimum separation of frequencies.

**Time**: March 18, 2015, 3:10 pm, SC 1431

**Speaker**: Qiang Wu, Middle Tennessee State University

**Title**: Mathematical Foundation of the Minimum Error Entropy Algorithm

**Abstract**: Information theoretical learning (ITL) is an important research area in signal processing and machine learning. It uses concepts of entropies and divergences from information theory to substitute the conventional statistical descriptors of variances and covariances. The empirical minimum error entropy (MEE) algorithm is a typical approach falling into this this framework and has been successfully used in both regression and classification problems.

In this talk, I will discuss the consistency analysis of the MEE algorithm. For this purpose, we introduce two types of consistency. The error entropy consistency, which requires the error entropy of the learned function to approximate the minimum error entropy, is proven when the bandwidth parameter tends to 0 at an appropriate rate. The regression consistency, which requires the learned function to approximate the regression function, however, is a complicated issue. We prove that the error entropy consistency implies the regression consistency for homoskedastic models where the noise is independent of the input variable. But for heteroskedastic models, a counterexample is constructed to show that the two types of consistency are not necessarily coincident. A surprising result is that the regression consistency holds when the bandwidth parameter is sufficiently large. Regression consistency of two classes of special models is shown to hold with fixed bandwidth parameter. These results illustrate the complication of the MEE algorithm.

**Time**: January 23, 2015 (Friday). 3:10 pm, SC 1431

**Speaker**: Shahaf Nitzan, Kent State University

**Title**: Exponential frames on unbounded sets

**Abstract**: In contrast to orthonormal and Riesz bases, exponential frames (i.e., ‘over complete bases’) are in many cases easy to come by. In particular, it is not difficult to show that every bounded set of positive measure admits an exponential frame. When unbounded sets (of finite measure) are considered, the problem becomes more delicate. In this talk I will discuss a joint work with A. Olevskii and A. Ulanovskii, where we prove that every such set admits an exponential frame. To obtain this result we apply one of the outcomes of Marcus, Spielman and Srivastava’s recent solution of the Kadison-Singer conjecture. This talk is part of the Shanks Workshop on “Uncertainty Principles in Time Frequency Analysis”

## 2014

**Time**: November 12, 2014. 3:10 pm, SC 1432

**Speaker**: Maryke van der Walt, University of Missouri, St. Louis

**Title**: Signal analysis via instantaneous frequency estimation of signal components

**Abstract**: The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang et al in 1998, is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF’s), after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu and H.-T. Wu in 2011, is applied to estimate the IF’s of all signal components of the given signal, based on one single reference “IF function,” under the assumption that the signal components satisfy certain strict properties of a so-called adaptive harmonic model (AHM), before the signal components of the model are recovered. The objective of our paper is to develop a hybrid EMD-SST computational scheme by applying a “modified SST” to each IMF of the EMD, as an alternative approach to the original EMD-HSA method. While our modified SST assures non-negative instantaneous frequencies of the IMF’s, application of the EMD scheme eliminates the dependence of a single reference IF value as well as the guessing work of the number of signal components in the original SST approach. Our modification of the SST consists of applying vanishing moment wavelets (introduced in a recent paper by C.K. Chui and H.-T. Wu) with stacked knots to process signals on bounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection through generalized cross-validation. In addition, we formulate a local cubic spline interpolation scheme for real-time realization of the EMD sifting process that improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite time intervals. This is a joint work with C.K. Chui.

**Time**: November 5, 2014. 3:10 pm, SC 1432

**Speaker**: Guilherme de Silva, KU Leuven

**Title**: Breaking the Symmetry in the Normal Matrix Model

**Abstract**: We consider the normal matrix model with cubic plus linear potential. The model is ill-defined, and to regualrize it, Elbau and Felder proposed to make a cut-off on the complex plane, leading to a system of orthogonal polynomials with respect to a certain 2D measure. When studying this model with a monic cubic weight, Bleher and Kuijlaars associated to this model a system of non-hermitian multiple orthogonal polynomials, which are expected to be asymptotically the same as the 2D orthogonal polynomials

In this talk, we will focus on the non-hermitian MOP’s in the spirit of Bleher and Kuijlaars, but now adding a linear term to the cubic potential. It will be shown how some quantities of the normal matrix model are related to those orthogonal polynomials. At the technical level, the linear term breaks the symmetry of the model, and in order to deal with it, we introduce a quadratic differential on the spectral curve and describe globally its trajectories. The trajectories of the quadratic differential play a fundamental role in the asymptotic analysis of the MOP’s.

This is an ongoing project with Pavel Bleher (Indiana University – Purdue University Indianapolis).

**Time**: October 1, 2014. 3:10 pm, SC 1432

**Speaker**: Andrei Martinez-Finkelshtein, University of Almeria (visiting Vanderbilt)

**Title**: Two approximation problems in ophthalmology, or how Gauss can beat Zernike

**Abstract**: Modern corneal topographers or videokeratometers based on the principle of Placido disks collect the data (either corneal altimetry or corneal power) in a discrete set of points on the disk organized in a nearly concentric pattern. A reliable reconstruction of the cornea from this information is essential for a correct early diagnosis of several ophthalmological diseases. A standard procedure used in clinical practice is based on a least squares fit by Zernike polynomials (an orthonormal family with respect to the plane measure on the disk). However well this method works for regular corneas, it has several drawbacks and lacks precision in more complex (and thus, clinically relevant) cases. On the other hand, the point-spread-function (PSF) of an eye carries important information about the eye as an optical instrument. PSF can be found from non-invasive objective measurements, e.g. from the wavefront aberrations of the eye. However, the actual calculation of the PSF (which boils down to computing 2D Fourier transforms of functions on a disk for different parameters) is costly. Here also the Zernike polynomials play a predominant role, laying the groundwork for the so-called Extended Nijboer-Zernike analysis. It turns out that in both problems the gaussian functions can be used as an alternative to Zernike polynomials. For the first problem, we devise an adaptive and multi-scale algorithm that fits the corneal data by means of anisotropic Gaussian radial basis functions. The shape parameters of these functions, chosen dynamically in dependence of the data, constitute an important additional source of information about the corneal irregularity. For the second problem, an approximation of the wavefront aberrations by gaussian functions results in a fast and reliable method of parallel computation of these 2D Fourier integrals and of the through-focus characteristics of a human eye.

**Time**: September 24, 2014. 3:10 pm, SC 1432

**Speaker**: Dustin Mixon, Air Force Institute of Technology

**Title**: Phase retrieval: Approaching the theoretical limits in practice

**Abstract**: In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. Very little is known about how to design injective intensity measurements, let alone stable measurements with efficient reconstruction algorithms. This talk helps to fill the void – I will discuss a wide variety of recent results in phase retrieval, including various conditions for injectivity and stability (joint work with Afonso S. Bandeira (Princeton), Jameson Cahill (Duke) and Aaron A. Nelson (AFIT)) as well as measurement designs based on spectral graph theory which allow for efficient reconstruction (joint work with Boris Alexeev (Princeton), Afonso S. Bandeira (Princeton) and Matthew Fickus (AFIT)). In particular, I will show how Fourier-type tricks can be leveraged in concert with this graph-theoretic design to produce pseudorandom aperatures for X-ray crystallography and related disciplines (joint work with Afonso S. Bandeira (Princeton) and Yutong Chen (Princeton)).

**Time**: April 2, 2014. 3:10 pm, SC 1307

**Speaker**: Anne Gelb, Arizona State University

**Title**: Numerical Approximation Methods for Non-Uniform Fourier Data

**Abstract**: In this talk I discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR). Two standard reconstruction techniques, convolutional gridding (the non-uniform FFT) and uniform resampling, are summarized, and some of the difficulties are discussed. It is then demonstrated how spectral reprojection can be used to mollify both the Gibbs phenomenon and the error due to the non-uniform sampling. It is further shown that incorporating prior information, such as the internal edges of the underlying function, can greatly improve the reconstruction quality. Finally, an alternative approach to the problem that uses Fourier frames is proposed.

**Time**: February 12, 2014. 3:10 pm, SC 1307

**Speaker**: Charles Martin, Vandebilt University

**Title**: Perturbations of Green Functions and the Dirichlet Problem

**Abstract**: The Dirichlet problem for the Laplacian on a domain is better understood and more easily computed than it is for that of a more general elliptic operator. If an elliptic operator is somehow a small perturbation from the Laplacian, what corrections can we make to the solutions to the Dirichlet problem? In this talk we’ll address this question by first considering perturbation of Green functions. With various perturbative formulas (and a few series expansions) in hand, we turn to the problem of bounding the resulting error terms.

**Time**: January 22, 2014. 3:10 pm, SC 1307

**Speaker**: Stefano de Marchi, University of Padua

**Title**: Padua points: theory, computation, applications and open problems.

**Abstract**: The so called “Padua points” are the first set of unisolvent points in the square that give a simple, geometric, and explicit construction of bivariate polynomial interpolation. Their associated Lebesgue constant, which measures the goodness of approximation, has minimal order of growth, i.e. O(log^2(n)) with n the polynomial degree. In the talk we present a stable and efficient implementation of the corresponding Lagrange interpolation and quadrature formulas. We also discuss extensions of (non-polynomial) Padua-like interpolation to other domains, such as triangles and ellipses. Applications to finding approximate Fekete points on tensor-product domains are also discussed. We conclude with some open problems.

## 2013

**Time**: November 20, 2013. 3:10 pm, SC 1307

**Speaker**: Igor Pritsker, Oklahoma State University

**Title**: Riesz decomposition for the farthest distance functions via logarithmic, Green and Riesz potentials.

**Abstract**: We discuss several versions of the Riesz Decomposition Theorem for superharmonic functions. This theorem is usually stated for Newtonian and logarithmic potentials in the literature, but it isalso true for some Riesz kernels. However, no full version for Riesz potentials is available. We mention related topics on $\alpha$-superharmonic and polyharmonic functions, and on fractional Laplacian. We apply Riesz decompositions to obtain integral representations of the farthest distance functions for compact sets as logarithmic, Green and Riesz potentials of positive measures with unbounded support. The representing measures encode many geometric properties of compact sets via their distance functions.

**Time**: November 6, 2013. 3:10 pm, SC 1307

**Speaker**: Koushik Ramachandran, Purdue University

**Title**: Asymptotic behavior of positive harmonic functions in certain unbounded domains

**Abstract**: We derive asymptotic estimates at infinity for positive harmonic functions in large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone. Examples of such domains are various paraboloids and, horn domains.

**Time**: October 30, 2013. 3:10 pm, SC 1307

**Speaker**: Mark Iwen, Michigan State University

**Title**: Fast Algorithms for Fitting High-Dimensional Data with Hyperplanes

**Abstract**: I will discuss computational methods for fitting large sets of points in high dimensional Euclidean space with low-dimensional subspaces that are “near-optimal”. Several different measures of optimality will be considered, including one closely related to kolmogorov n-widths. In this last setting we will present a fast (i.e., linear time in the number of points) algorithm with rigorous approximation guarantees.

**Time**: October 9, 2013. 3:10 pm, SC 1307

**Speaker**: Jorge Roman, Vanderbilt University

**Title**: An Introduction to Markov Chain Monte Carlo Methods

**Abstract**: The need to approximate an intractable integral with respect to a probability distribution P is a problem that frequently arises across many different disciplines. A popular alternative to numerical integration and analytical approximation methods is the Monte Carlo (MC) method which uses computer simulations to estimate the integral. In the MC method, one generates independent and identically distributed (iid) samples from P and then uses sample averages to estimate the integral. However, in many situations, P is a complex high-dimensional probability distribution and obtaining iid samples from it is either impossible or impractical. When this happens, one may still be able to use the increasingly popular Markov chain Monte Carlo (MCMC) method in which the iid draws are replaced by a Markov chain that has P as its stationary distribution. In this talk, I will give a brief introduction to the MC and MCMC methods. The focus will be on the MCMC method and its applications to Bayesian statistics.

**Time**: October 2, 2013. 3:10 pm, SC 1307

**Speaker**: Ding-Xuan Zhou, City University Hong Kong

**Title**: Learning Theory and Minimum Error Entropy Principle

**Abstract**:

**Time**: September 25, 2013. 3:10 pm, SC 1307

**Speaker**: Jean-Luc Bouchot, Drexel University

**Title**: Progress on Hard Thresholding Pursuit

**Abstract**: The Hard Thresholding Pursuit algorithm for sparse recovery is revisited using a new theoretical analysis. The main result states that all sparse vectors can be exactly recovered from incomplete linear measurements in a number of iterations at most proportional to the sparsity level as soon as the measurement matrix obeys a restricted isometry condition. The recovery is also robust to measurement error The same conclusions are derived for a variation of Hard Thresholding Pursuit, called Graded Hard Thresholding Pursuit, which is a natural companion to Orthogonal Matching Pursuit and runs without a prior estimation of the sparsity level. In two extreme cases of the vector shape, it is also shown that, with high probability on the draw of random measurements, a fixed sparse vector is robustly recovered in a number of iterations precisely equal to the sparsity level. These theoretical findings are experimentally validated, too.

**Time**: September 18, 2013. 3:10 pm, SC 1307

**Speaker**: Matt Fickus, Air Force Institute of Technology

**Title**: Compressed Sensing with Equiangular Tight Frames

**Abstract**: Compressed sensing (CS) is changing the way we think about measuring high-dimensional signals and images. In particular, CS promises to revolutionize hyperspectral imaging. Indeed, emerging camera prototypes are exploiting random masks in order to greatly reduce the exposure times needed to form hyperspectral images. Here, the randomness of the masks is due to the crucial role that random matrices play in CS. In short, in terms of CS’s restricted isometry property (RIP), random matrices far outshine all known deterministic matrix constructions. To be clear, for most deterministic constructions, it is unknown whether this performance shortfall (known as the “square-root bottleneck”) is simply a consequence of poor proof techniques or, more seriously, a flaw in the matrix design itself. In the remainder of this talk, we focus on this particular question in the special case of matrices formed from equiangular tight frames (ETFs). ETFs are overcomplete collections of unit vectors with minimal coherence, namely optimal packings of a given number of lines in a Euclidean space of a given dimension. We discuss the degree to which the recently-introduced Steiner and Kirkman ETFs satisfy the RIP. We further discuss how a popular family of ETFs, namely harmonic ETFs arising from McFarland difference sets, are particular examples of Kirkman ETFs. Overall, we find that many families of ETFs are shockingly bad when it comes to RIP, being provably incapable of exceeding the square-root bottleneck. Such ETFs are nevertheless useful in variety of other real-world applications, including waveform design for wireless communication and algebraic coding theory.

**Time**: August 28, 2013. 3:10 pm, SC 1307

**Speaker**: Oleg Davydov, Strathclyde University (Scotland)

**Title**: Error bounds for kernel-based numerical differentiation

**Abstract**: The literature on meshless methods observed that kernel-based numerical differentiation formulae are highly accurate and robust. We present error bounds for such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds for kernel-based interpolation but are not applicable in this setting. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we show that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. The talk is based on joint research with Robert Schaback.

**Time**: April 10, 2013. 3:10 pm, SC 1307

**Speaker**: Maria Navascues, University of Zaragoza

**Title**: Some historical precedents of fractal functions

**Abstract**: In this talk, we wish to pay tribute to the scientists of older generations, who, through their reseatch, lead to the current state of knowledge of the fractal functions. We review the fundamental milestones of the origin and evolution of the self-similar curves that, in some cases, agree with continuous and nowhere differentiable functions, but they are not exhausted by them. Our main interest is to emphasize the lesser known examples, due to a deficient or late publication (Bolzano’s map for instance).

We will review different ways of defining self-similar curves. We will recall the first functions without tangent, but also some fractal functions having derivative almost everywhere. Most of the models studied may seem quite paradoxical (“monsters” in the words of Poincare) as, for instance, curves with a fractal dimension of two and having a tangent at every point. These instances suggest that the classification and even the definition of fractal functions are far from being established. The strategies of definition of each example compose a toolbox that will provide the audience with a selection of procedures for the construction of its own fractal function.

**Time**: April 3, 2013. 3:10 pm, SC 1307

**Speaker**: Keri Kornelson, University of Oklahoma

**Title**: Fourier bases on fractals

**Abstract**: The study of Bernoulli convolution measures dates back to the 1930’s, yet there has been a recent resurgence in the theory prompted by the connection between convolution measures and iterated function systems (IFSs). The measures are supported on fractal Cantor subsets of the real line, and exhibit their own sort of self-similarity. We will use the IFS connection to discover Fourier bases on the L^2 Hilbert spaces with respect to Bernoulli convolution measures.

There are some interesting phenomena that arise in this setting. We find that some Cantor sets support Fourier bases while others do not. In cases where a Fourier basis does exist, we can sometimes scale or shift the Fourier frequencies by an integer to obtain another ONB. We also discover properties of the unitary operator mapping between two such bases. The self-similarity of the measure and the support space can, in some cases, carry over into a self-similarity of the operator.

**Time**: March 27, 2013. 3:10 pm, SC 1307

**Speaker**: Johan De Villiers, Stellenbosch University

**Title**: Wavelet Analysis Based on Algebraic Polynomial Identities

**Abstract**: By starting out from a given refinable function, and relying on a corresponding space decomposition which is not necessarily orthogonal, we present a general wavelet construction method based on the solution of a system of algebraic polynomial identities. The resulting decomposition sequences are finite, and, for any given vanishing moment order, the wavelets thus constructed are minimally supported, and possess robust- stable integer shifts. The special case of cardinal B-splines are given special attention.

**Time**: February 20, 2013. 3:10 pm, SC 1307 (cancelled)

**Speaker**: Kamen Ivanov, University of South Carolina

**Title**: TBA

**Abstract**: TBA

**Time**: February 13, 2013. 3:10 pm, SC 1307

**Speaker**: Roza Aceska, Vanderbilt University

**Title**: Gabor frames, Wilson bases and multi-systems

**Abstract**: Frames can be seen as generalized bases, that is, over-complete collections, which are used for stable representations of signals as linear combinations of basic building atoms. It is very useful when we can use locally adapted atoms, which in addition behave as elements of local bases. We explore the possibility of using localized parts of frames and bases when building a customized frame. After a review on Gabor frames and Wilson bases, we consider the question of combining parts of these collections into a multi-frame set and look at its properties.

**Time**: February 7, 2013. 4:10 pm, SC 1425 (also a Colloquium)

**Speaker**: Barry Simon, Caltech

**Title**: Tales of Our Forefathers

**Abstract**: This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I’ll convince you they were also human beings and that, as the Chinese say, “May you live in interesting times” really is a curse.

**Time**: January 30, 2013. 3:10 pm, SC 1307

**Speaker**: Eduardo Lima (MIT) and Laurent Baratchart (INRIA)

**Title**: Overview of Inverse Problems in Planar Magnetization

**Abstract**: TBA

## 2012

**Time**: November 28, 2012. 3:10 pm, SC 1307

**Speaker**: Manos Papadakis, University of Houston

**Title**: Texture Analysis in 3D for the Detection of Liver Cancer in X-ray CT Scans

**Abstract**: We propose a method for the 3D-rigid motion invariant texture discrimination for discrete 3D-textures that are spatially homogeneous. We model these textures as stationary Gaussian random fields. We formally develop the concept of 3D-texture rotations in the 3D-digital domain. We use this novel concept to define a `distance’ between 3D-textures that remains invariant under all 3D-rigid motions of the texture. This concept of `distance’ can be used for a monoscale or a multiscale setting to test the 3D-rigid motion invariant statistical similarity of stochastic 3D-textures. To extract this novel texture `distance’ we use the Isotropic Mutliresolution Analysis. We also show how to construct wavelets associated with this structure by means of extension principles and we discuss some very recent results by Atreas, Melas and Stavropoulos on the geometric structure underlying the various extension principles. The 3D-texture `distance’ is used to define a set of 3D-rigid motion invariant texture features. We experimentally establish that when they are combined with mean attenuation intensity differences the new augmented features are capable of discriminating normal from abnormal liver tissue in arterial phase contrast enhanced X-ray CT-scans with high sensitivity and specificity. To extract these features CT-scans are processed in their native dimensionality. We experimentally observe that the 3D-rotational invariance of the proposed features improves the clustering of the feature vectors extracted from normal liver tissue samples. This work is joint with R. Azencott, S. Jain, S. Upadhyay, I.A. Kakadiaris and G. Gladish, MD.

**Time**: November 14, 2012. 3:10 pm, SC 1307

**Speaker**: Ben Adcock, Purdue University

**Title**: Breaking the coherence barrier: semi-random sampling in compressed sensing

**Abstract**: Compressed sensing is a recent development in the field of sampling Based on the notion of sparsity, it provides a theory and techniques for the recovery of images and signals from only a relatively small number of measurements. The key ingredients that permit this so-called subsampling are (i) sparsity of the signal in a particular basis and (ii) mutual incoherence between such basis and the sampling system. Provided the corresponding coherence parameter is sufficiently small, one can recover a sparse signal using a number of measurements that is, up to a log factor, on the order of the sparsity.

Unfortunately, many problems that one encounters in practice are not incoherent. For example, Fourier sampling, the type of sampling encountered in Magnetic Resonance Imaging (MRI), is typically not incoherent with wavelet or polynomials bases. To overcome this `coherence barrier’ we introduce a new theory of compressed sensing, based on so-called asymptotic incoherence and asymptotic sparsity. When combined with a semi-random sampling strategy, this allows for significant subsampling in problems for which standard compressed sensing tools are limited by the lack of incoherence. Moreover, we demonstrate how the amount of subsampling possible with this new approach actually increases with resolution. In other words, this technique is particularly well suited to higher resolution problems.

This is joint work with Anders Hansen and Bogdan Roman (University of Cambridge)

**Time**: TBA (postponed from October 31)

**Speaker**: Doron Lubinsky, Georgia Institute of Technology

**Title**: L^{p} Christoffel functions and Paley-Wiener spaces

**Abstract**: Let ω be a finite positive Borel measure on the unit circle. Let p>0 and

λ _{n,p}(ω,z) =inf_{deg P ≤ n-1} (∫_{-π}^{π}|P(e^{iθ})| ^{p}dω(θ))(|P(z)| ^{p})^{-1}denote the corresponding L_{p} Christoffel function. The asymptotic behavior of λ_{n,p}(ω,z) as n→∞ is well understood for |z|<1, falling naturally within the ambit of Szego theory. We provide asymptotics on the unit circle, for all p>0. These involve an extremal problem for L_{π}^{p}, the Paley-Wiener space of entire functions f of exponential type at most π, with ∫_{-∞}^{∞}|f|^{p}<∞. Let

E_{p}=inf {∫_{-∞}^{∞}| f|^{p} : f∈ L_{π}^{p} with f(0) =1}.We show that for all p>0,

lim_{n→∞}nλ_{n,p}(ω,z)=2π E_{p}ω^{‘}(z) ,when ω is a regular measure on the unit circle, and z is a Lebesgue point of ω, while ω^{‘} is lower semi-continuous at z. For p≠2, they seem to be new even for Lebesgue measure on the unit circle. In addition, for p>1, we establish universality type limits. Let P_{n,p,z} be a polynomial of degree at most n-1 with P_{n,p,z}( z)=1, attaining the infimum above. We prove that uniformly for u in compact subsets of the plane,

lim_{n→∞}P_{n,p,z}(ze^{2πiu/n})=e^{iuπ }f_{p}(u)where f_{p}∈ L_{π}^{p} satisfies f_{p}(0)=1 and attains the second infimum in above. When p=2, this reduces to a special case of the universality limit associated with random matrices. Analogous results are presented for measures on [-1,1].

**Time**: October 17, 2012. 3:10 pm, SC 1307

**Speaker**: Matt Hirn, Yale University

**Title**: Diffusion maps for changing data

**Abstract**: Much of the data collected today is massive and high dimensional, yet hidden within is a low dimensional structure that is key to understanding it. As such, recently there has been a large class of research that utilizes nonlinear mappings into low dimensional spaces in order to organize high dimensional data according to its intrinsic geometry. Examples include, but are not limited to, locally linear embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps. The type of question we shall ask in this talk is the following: if my data is in some way dynamic, either evolving over time or changing depending on some set of input parameters, how do these low dimensional embeddings behave? Is there a way to go between the embeddings, or better still, track the evolution of the data in its intrinsic geometry? Can we understand the global behavior of the data in a concise way? Focusing on the diffusion maps framework, we shall address these questions and a few others. We will begin with a review the original work on diffusion maps by Coifman and Lafon, and then present some current theoretical results. Various synthetic and real world examples will be presented to illustrate these ideas in practice, including examples taken from image analysis and dynamical systems. Parts of this talk are based on joint work with Ronald Coifman, Simon Adar, Yoel Shkolnisky, Eyal Ben Dor, and Roy Lederman.

**Time**: October 10, 2012. 3:10 pm, SC 1307

**Speaker**: Yaniv Plan, University of Michigan

**Title**: One-bit matrix completion

**Abstract**: Let Y be a matrix representing voting results in which each entry is either 1 or -1. For example, we may take Y_{ij} = 1 if senator i votes “yes” on bill j, and -1 otherwise. Now suppose that a number of entries are missing from Y (for example, senators may be out of town during a vote). Could you guess how to fill in the missing entries (how would senator i have voted on bill j)? Similar questions arise in many other applications such as recommender systems or binary survey completion. In this talk, we assume that the binary data is generated according a probability distribution which is parameterized by an underlying matrix M. Further, we assume that M has low rank – loosely, this means that the voting preferences of each senator may be defined by just a few characteristics (Democrat, Republican, etc.), although these characteristics need not be known. We show that the probability distribution of the missing entries of Y may be well approximated using maximum likelihood estimation under a nuclear-norm constraint. Under appropriate assumptions, we demonstrate that the approximation error is nearly minimax. The upper bounds are proven using techniques from probability in Banach spaces. The lower bounds are proven using information theoretic techniques, in particular Fano’s inequality.

**Time**: September 26, 2012. 3:10 pm, SC 1307

**Speaker**: Hau-tieng Wu, University of California Berkeley

**Title**: Instantaneous frequency, shape functions, Synchrosqueezing transform, and some applications

**Abstract**: PDF

**Time**: September 5, 2012. 3:10 pm, SC 1307

**Speaker**: Maxim Yattselev, University of Oregon

**Title**: Bernstein-Szego Theorem on Algebraic S-Contours

**Abstract**: PDF

**Time**: April 25, 2012. 3:10 pm, SC1310

**Speaker**: Antoine Ayache, Laboratoire Paul Painlevé

**Title**: Optimal Series Representations of Continuous Gaussian Random Fields

**Abstract**: Any continuous Gaussian random field X(t) can be represented as a weighted combination (with weights a sequence of independent standard Gaussian random variables) of a sequence of deterministic continuous functions that is almost surely convergent in a Banach space of continuous functions. A representation of this type is said to be optimal when the norm of the tail of the series converges to zero as fast as possible. X(t) is associated to a sequence of “l-numbers”, which determine this fastest possible rate, and the asymptotic behavior of the latter sequence can be estimated by using operator theory; also, it is worth noticing that the latter behavior is closely connected with the behavior of small ball probabilities of {X(t)}t?[0,1]N. The main three goals of our talk are the following: (a) to connect the Holder regularity of {X(t)}t?[0,1]N with the rate of convergence of its l-numbers; (b) to show that the Meyer, Sellan and Taqqu wavelet series representations of fractional Brownian motion are optimal; (c) to investigate, for the Riemann-Liouville process (that is the high frequency part of fractional Brownian motion), the optimality of the series representations obtained via the Haar and the trigonometric systems.

**Time**: April 18, 2012. 3:10 pm, SC1310

**Speaker**: Rayan Saab, Duke University

**Title**: High Accuracy Finite Frame Quantization Using Sigma-Delta Schemes

**Abstract**: In this talk, we address the digitization of oversampled signals in the finite-dimensional setting. In particular, we show that by quantizing the $N$-dimensional frame coefficients of signals in $\R^d$ using Sigma-Delta quantization schemes, it is possible to achieve root- exponential accuracy in the oversampling rate $\lambda:= N/d$ (even when one bit per measurement is used). These are currently the best known error rates in this context. To that end, we construct a family of finite frames tailored specifically for Sigma-Delta quantization. Our construction allows for error guarantees that behave as $e^{-c\sqrt{\lambda}}$, where under a mild restriction on the oversampling rate, the constants are absolute. Moreover, we show that harmonic frames can be used to achieve the same guarantees, but with the constants now depending on d. Finally, we show a somewhat surprising result whereby random frames achieve similar, albeit slightly weaker guarantees (with high probability). Finally, we discuss connections to quantization of compressed sensing measurements. This is joint work, in part with F. Krahmer and R. Ward, and in part with O. Yilmaz.

**Time**: April 11, 2012. 3:10 pm, SC1310

**Speaker**: Pete Casazza, University of Missouri

**Title**: Algorithms for Threat Detection

**Abstract**: Fusion frames are a recent development in Hilbert space theory which have broad application to modeling problems in distributed processing, visual/hearing systems, geophones in geophysics, forest fire detection and much more. We will look at recent applications of fusion frames to wireless sensor networks for detecting and intercepting chemical/biological/nuclear materials which are being transported. This is a totally new subject and so we will present many more problems than solutions.

**Time**: January 25, 2012. 3:10 pm, SC1310

**Speaker**: Anthony Mays, University of Melbourne

**Title**: A Geometrical Triumvirate of Random Matrices

**Abstract**: A random matrix is, broadly speaking, a matrix with entries randomlychosen from some distribution. In the non-random case eigenvalues canoccur anywhere in the complex plane, but, remarkably, random elements imply predictable behaviour, albeit in a probabilistic sense. Correlation functions are one measure of a probabilistic characterisation and we discuss a 5-part scheme, based upon orthogonal polynomials, to calculate the eigenvalue correlation functions. We apply this scheme to three ensembles of random matrices, each of which can be identified with one of the surfaces of constant Gaussian curvature: the plane, the sphere and the anti- or pseudo-sphere. We will be using real random matrices, which possess the added complication of having a finite probability of real eigenvalues. This talk aims to be accessible, and to that end we will start with a general overview of random matrices and then discuss the 5-step method, hopefully keeping technicalities to a minimum, and with plenty of pictures.

## 2011

**Time**: October 26, 2011. 3:10 pm, SC1310

**Speaker**: Xuemei Chen, Vanderbilt University

**Title**: Almost Sure Convergence for the Kaczmarz Algorithm with Random Measurements

**Abstract**: The Kaczmarz algorithm is an iterative method for reconstructing a signal $x\in\R^d$ from an overcomplete collection of linear measurements $y_n = \langle x, \varphi_n \rangle$, $n \geq 1$. We prove quantitative bounds on the rate of almost sure exponential convergence in the Kaczmarz algorithm for suitable classes of random measurement vectors $\{\varphi_n\}_{n=1}^{\infty} \subset \R^d$. Refined convergence results are given for the special case when each $\varphi_n$ has i.i.d. Gaussian entries and, more generally, when each $\varphi_n/\|\varphi_n\|$ is uniformly distributed on $\mathbb{S}^{d-1}$. This work on almost sure convergence complements the mean squared error analysis of Strohmer and Vershynin for randomized versions of the Kaczmarz algorithm.

**Time**: October 12, 2011. 3:10 pm, SC1310

**Speaker**: Baili Min, Vanderbilt University

**Title**: Approach Regions for Domains in $\CC^2$ of Finite Type

**Abstract**: Recall the Fatou theorem for the unit disc in $\CC$. In this talk we will see the generalization to the domain in $\CC^2$. More exactly, we will see strongly pseudoconvex domains and those of finite type. We are also going to show that the approach regions studied by Nagel, Stein, Wainger and Neff are the best possible ones for the boundary behavior of bounded analytic functions, and there is no Fatou theorem for complex tangentially broader approach regions.

**Time**: October 5, 2011. 3:10 pm, SC1310

**Speaker**: J. Tyler Whitehouse, Vanderbilt University

**Title**: Consistent Reconstruction and Random Polytopes

**Time**: September 14, 2011. 3:10 pm, SC1310

**Speaker**: Aleks Ignjatovic, University of New South Wales

**Title**: Chromatic Derivatives and Approximations

**Abstract**: Chromatic derivatives are special, numerically robust linear differential operators which provide a unification framework for a broad class of orthogonal polynomials with a broad class of special functions. They are used to define chromatic expansions which generalize the Neumann series of Bessel functions. Such expansions are motivated by signal processing; they provide local signal representation complementary to the global signal representation given by the Shannon sampling expansion. They were introduced for the purpose of designing a switch mode amplifier. Unlike the Taylor expansion which they are intended to replace, they share all the properties of the Shannon expansion which are crucial for signal processing. Besides being a promissing new tool for signal processing, chromatic derivatives and expansions have intriguing mathematical properties related to harmonic analysis. For example, they naturaly introduce spaces of almost periodic functions which corespond to orthogonal polynomials of a very broad class, containing classical families of orthogonal polynomials. We will alo present an open conjecture related to a possible generalization of the Paley Wiener Theorem.

**Time**: September 21, 2011. 3:10 pm, SC1310

**Speaker**: Aleks Ignjatovic, University of New South Wales

**Title**: Chromatic Derivatives and Approximations (Continued)

**Abstract**: Chromatic derivatives are special, numerically robust linear differential operators which provide a unification framework for a broad class of orthogonal polynomials with a broad class of special functions. They are used to define chromatic expansions which generalize the Neumann series of Bessel functions. Such expansions are motivated by signal processing; they provide local signal representation complementary to the global signal representation given by the Shannon sampling expansion. They were introduced for the purpose of designing a switch mode amplifier. Unlike the Taylor expansion which they are intended to replace, they share all the properties of the Shannon expansion which are crucial for signal processing. Besides being a promissing new tool for signal processing, chromatic derivatives and expansions have intriguing mathematical properties related to harmonic analysis. For example, they naturaly introduce spaces of almost periodic functions which corespond to orthogonal polynomials of a very broad class, containing classical families of orthogonal polynomials. We will alo present an open conjecture related to a possible generalization of the Paley Wiener Theorem.

**Time**: April 13, 2011. 4:10 pm, SC1312.

**Speaker**: Hans-Peter Blatt, Katholische University Eichstatt

**Title**: Growth behavior and value distibution of rational approximants

**Abstract**: We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree n and denominator degree m(n) for meromorphic functions f on a compact set E of the complex plane, where m(n) = o(n/log n) as n tends to infinity. We obtain a Jentzsch-Szegö type result, i. e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain of meromorphy of f if the function f has a singularity of multivalued character on the boundary of this domain. In the case that f has an essential singularity on this domain, we can prove that such a point is an accumulation point of zeros or poles of best uniform rational approximants. An outlook is given for the approximation of f on an interval, where the function is not holomorphic. Applications for Padé approximation are discussed.

**Time**: February 23, 2011. 4:10 pm, SC1312.

**Speaker**: Thomas Hangelbroek, Vanderbilt University

**Title**: Boundary effects in kernel approximation and the polyharmonic Dirichlet problem

**Abstract**: In this talk I will discuss boundary effects in kernel approximation — specifically the pathology of the boundary as it relates to convergence rates. Accompanying this I will introduce a new approximation scheme, one that delivers theoretically optimal and previously unrealized convergence rates by isolating the boundary effects in easily managed integrals. Driving this is a potential theoretic integral representation derived from the boundary layer potential solution of the polyharmonic Dirichlet problem.

## 2010

**Time**: September 29, 2010. 4:10 pm, SC1312.

**Speaker**: Thomas Hangelbroek, Vanderbilt University

**Title**: Approximation and interpolation on Riemannian manifolds with kernels

**Abstract**: In this talk I will present very recent results for interpolation and approximation on compact Riemannian manifolds using kernels. I will introduce a new family of kernels and discuss the rapid decay of associated Lagrange functions, the Lp stability of bases for the underlying kernel spaces, the uniform boundedness of Lebesgue constants, the uniform boundedness of the L2 projector in Lp, and progress on specific problems on spheres and SO(3). If time permits, I’ll discuss how such kernels can be used to treat highly non-uniform arrangements of data.

**Time**: September 15, 2010. 4:10 pm, SC1312.

**Speaker**: Dominik Schmid, Institute of Biomathematics and Biometry at the German Research Center for Environmental Health

**Title**: Approximation on the rotation group

**Abstract**: Scattered data approximation problems on the rotation group naturally arise in various fields in science in engineering. After introducing such problems, we briefly present different approaches to handle such questions. By considering one of these approaches in more detail, we will encounter so-called Marcinkiewicz-Zygmund inequalities. These inequalities provide a norm equivalence between the continuous and discrete $L^p$- norm of certain basis functions and are a very powerful tool in order to answer important questions that come along with the approximation of scattered data on the underlying structure. We will present the main tools and techniques that enable us to establish such inequalities in our setting of the rotation group.

**Time**: April 30, 2010. 4:10 pm, room TBA.

**Speaker**: Hendrik Speleers, Catholic University of Leuven

**Title**: Convex splines over triangulations

**Abstract**: Convexity is often required in the design of surfaces. Typically, a nonlinear optimization problem arises, where the objective function controls the fairness of the surface and the constraints include convexity conditions. We consider convex polynomial spline functions defined on triangulations. In general, convexity conditions on polynomial patches are nonlinear. In order to simplify the optimization problem, it is advantageous to have linear conditions. We present a simple construction to generate sets of sufficient linear convexity conditions for polynomials defined on a triangle. This general approach subsumes the known sets of linear conditions in the literature. In addition, it allows us to give a geometric interpretation, and we can easily construct sets of linear conditions that are symmetric..

**Time**: April 27, 2010. 4:10 pm, room 1312.

**Speaker**: Abey Lopez, Vanderbilt University

**Title**: Multiple orthogonal polynomials on star like sets

**Abstract**: I will describe different asymptotic properties of multiple orthogonal polynomials associated with measures supported on a star centered at the origin with equidistant rays. The ratio asymptotic behavior can be described with the help of a certain compact Riemann surface of genus zero. The nth root asymptotic behavior and zero asymptotic distribution are described in terms of the solution to a vector equilibrium problem for logarithmic potentials. All the necessary definitions will be properly introduced. Some conjectures about the limiting behavior of the recurrence coefficients associated with these polynomials will be mentioned. This work complements recent investigations of Aptekarev, Kalyagin and Saff on strong asymptotics of monic polynomials generated by a three-term recurrence relation of arbitrary order..

**Time**: April 23, 2010. 3:10 pm, room 1310.

**Speaker**: Radu Balan, University of Maryland

**Title**: Signal Reconstruction From Its Spectrogram

**Abstract**: This paper presents a framework for discrete-time signal reconstruction from absolute values of its short-time Fourier coefficients. Our approach has two steps. In step one we reconstruct a band-diagonal matrix associated to the rank-one operator $K_x=xx^*$. In step two we recover the signal $x$ by solving an optimization problem. The two steps are somewhat independent, and one purpose of this talk is to present a framework that decouples the two problems. The solution to the first step is connected to the problem of constructing frames for spaces of Hilbert-Schmidt operators. The second step is somewhat more elusive. Due to inherent redundancy in recovering $x$ from its associated rank-one operator $K_x$, the reconstruction problem allows for imposing supplemental conditions. In this paper we make one such choice that yields a fast and robust reconstruction. However this choice may not necessarily be optimal in other situations. It is worth mentioning that this second step is related to the problem of finding a rank-one approximation to a matrix with missing data.

**Time**: April 20, 2010. 4:10 pm, room 1312.

**Speaker**: Bernhard Bodmann, University of Houston

**Title**: Combinatorics and complex equiangular tight frames

**Abstract**: Equiangular tight frames combine a curious mix of spectral and geometric properties that makes them a fascinating topic in matrix theory. Moreover, these frames turn out to be optimal for certain applications in signal communications. Seidel has pioneered the use of combinatorial constructions of such frames for real Hilbert spaces. In a recent work with Helen Elwood, we follow Seidel’s footsteps into a corresponding combinatorial characterization of complex equiangular tight frames. To this end, we relate the construction of such frames to Hermitian matrices with two eigenvalues which contain $p$th roots of unity. We deduce necessary conditions for the existence of complex Seidel matrices, under the assumption that $p$ is prime. Explicitly examining the necessary conditions for smallest values of $p$ rules out the existence of many such frames with a number of vectors less than 50. Nevertheless, there are examples, which we confirm by constructing examples.

**Time**: April 13, 2010. 3:10 pm, room 1310.

**Speaker**: Wojciech Czaja, University of Maryland

**Title**: Multispectral imaging techniques for mapping molecular processes within the human retina

**Abstract**: We developed multispectral noninvasive fluorescence imaging techniques of the human retina. This is done by means of modifying standard fundus cameras by adding selected interference filter sets. The resulting multispectral datasets are processed by novel dimension reduction and classification algorithms. These algorithms resulted from a combination of the theory of frames with state of the art kernel based dimension reduction methods. Examples of applications of these techniques include detection of photoproducts in early Age-related Macular Degeneration, or mapping and monitoring macular pigment distributions.

**Time**: March 15, 2010. 3:00 pm, room 1312.

**Speaker**: Simon Foucart, University Pierre et Marie Curie

**Title**: Gelfand widths, pre-Gaussian random matrices, and joint sparsity

**Abstract**: In this talk, we explore three topics in Compressive Sensing. For the first topic, we outline the role of Gelfand widths before presenting natural (i.e., based only on ideas from Compressive Sensing) arguments to derive sharp estimates for the Gelfand widths of $\ell_p$-balls in $\ell_q$ when $0 < p \le 1$ and $p < q \le 2$. For the second topic, we show how sparse recovery via $\ell_1$-minimization can be achieved with pre-Gaussian random matrices using the minimal (up to constants) number of measurements. For the third topic, we explain why joint-sparse recovery by mixed $\ell_{1,2}$-minimization is not uniformly better than separate recovery by $\ell_1$-minimization, thus extending the equivalence between real and complex null space properties.

**Time**: February 2, 2010. 4:10 pm, room 1312.

**Speaker**: Luis Daniel Abreu, CMUC, University of Coimbra Portugal

**Title**: Time-frequency analysis of Bergman-type spaces

**Abstract**: In this talk we will present a real variable approach to some spaces of area measure (Bergmann-type) in the plane and in the upper-half plane. Underlying this approach is the Gabor transform with Hermite functions and the wavelet transform with Laguerre functions.

We will show how our method leads to new results. Some of them would be out of reach using “pure” Complex Analysis and only recent advances in time-frequency analysis (e.g. the structure of Gabor frames) made it possible to prove them

1) New(?) orthogonal functions in two variables with respect to area measure.

2) Sampling and interpolation in Fock spaces of polyanalytic functions (this is connected to recent work of Gröchenig and Lyubarskii).

3) Beurling density conditions for sampling and interpolation in Bergmann-type spaces.

4) Necessary density conditions for wavelet frames with Laguerre functions.

## 2009

**Time**: April 21, 2009. 4:10 pm, room 1312.

**Speaker**: Deanna Needell, University of California at Davis

**Title**: Greedy Algorithms in Compressed Sensing

**Abstract**: Compressed sensing is a new and fast growing field of applied mathematics that addresses the shortcomings of conventional signal compression. Given a signal with few nonzero coordinates relative to its dimension, compressed sensing seeks to reconstruct the signal from few nonadaptive linear measurements. As work in this area developed, two major approaches to the problem emerged, each with its own set of advantages and disadvantages. The first approach, L1-Minimization, provided strong results, but lacked the speed of the second, the greedy approach. The greedy approach, while providing a fast runtime, lacked stability and uniform guarantees. This gap between the approaches led researchers to seek an algorithm that could provide the benefits of both. We bridged this gap and provided a breakthrough algorithm, called Regularized Orthogonal Matching Pursuit (ROMP). ROMP is the first algorithm to provide the stability and uniform guarantees similar to those of L1-Minimization, while providing speed as a greedy approach. After analyzing these results, we developed the algorithm Compressive Sampling Matching Pursuit (CoSaMP), which improved upon the guarantees of ROMP. CoSaMP is the first algorithm to have provably optimal guarantees in every important aspect. This talk will provide a brief introduction to the area of compressed sensing and a discussion of these two recent developments.

**Time**: April 16, 2009. 4:10 pm, room 1312.

**Speaker**: Johann S. Brauchart, Graz University of Technology

**Title**: On an external field problem on the sphere

**Abstract**: Consider an isolated charged sphere in the presence of an external field exerted by a point charge over the North Pole (or, more generally, a line charge on the polar axis). The model of interaction is that of the Riesz $s$-potential $1 / r^s$ with $d-2 < s < d$. (Here, $d+1$ is the dimension of the embedding space.) We present results from joint work with Peter Dragnev (IPFW) and Ed Saff concerning the weighted extremal measure solving this external field problem and its properties (support, representation, potential). Interesting phenomena occur in the case $s to d-2$. Essential ingredients are the signed equilibrium on a spherical cap associated with the given external field (i.e. the signed measure whose potential is constant everywhere on this spherical cap), the Mhaskar-Saff functional (which yields the aforementioned constant), and the technique of iterated balayage to single out the spherical cap whose signed equilibrium becomes the weighted extremal measure.

**Time**: April 7, 2009. 4:10 pm, room 1312.

**Speaker**: Brody Johnson, St. Louis University

**Title**: Finite-Dimensional Wavelet Systems on the Torus

**Abstract**: The literature is rich with respect to treatments of wavelet bases for the real line. Early in the development of this wavelet theory some authors also considered wavelet systems for the torus; however, there has been considerably less work in this direction. Here, we consider a notion of finite-dimensional wavelet systems on the torus which, in many ways, adapts the theory of multiresolution analysis from the line to the torus. The orthonormal wavelet systems produced with this approach will be shown to offer arbitrarily close approximation of square-integrable functions on the torus. The talk will include a brief introduction to wavelet theory on the line.

**Time**: March 31, 2009. 4:10 pm, room 1312.

**Speaker**: Guillermo Lopez Lagomasino, Universidad Carlos III de Madrid

**Title**: On a class of perfect systems

**Abstract**: In 1873, CH. Hermite published the paper “On the exponential function” where he proved the transcendence of the number e. This paper is considered to mark the origin of the analytic theory of numbers. Years later, around 1936, on the basis of the method used by Hermite for systems of exponential functions, K. Mahler introduced the notion of perfect systems of first and second type. These are systems of functions satisfying certain algebraic independence for any polynomial combination of them. Until recently, very few special cases of systems of functions were known to be perfect. In 1980, E. M. Nikishin introduced what is now called a Nikishin system. These are systems of Markov type functions generated by measures supported on the real line. He also proved normality for such systems of functions when the degrees of the polynomials in the polynomial combination are equal (a system is said to be perfect if it is normal for polynomials of arbitrary degree). On the basis of this the question was posed as to whether or not Nikishin systems are perfect. In this talk we give a positive answer to the question.

**Time**: March 24, 2009. 4:10 pm, room 1312.

**Speaker**: Peter Massopust, Technical University of Munich

**Title**: Complex B-Splines: Theme and Variations

**Abstract**: The concept of a complex B-spline is introduced and some of its properties are discussed. Particular emphasis is placed on an interesting relation to Dirichlet averages that allows the derivation of a generalized Hermite-Gennochi formula. Using ridge functions, an extension of univariate complex B-splines to the multivariate setting is given. In this context, double Dirichlet averages are employed to define and compute moments of multivariate complex B-splines. Applications of complex B-splines to certain statistical processes are presented. This is joint work with Brigitte Forster.

**Time**: March 10, 2009. 4:10 pm, room 1312.

**Speaker**: Burcin Oktay, Bahkesir University, Turkey

**Title**: Approximation by Some Extremal Polynomials over Complex Domains

**Download Abstract**

**Time**: February 24, 2009. 4:10 pm, room 1312.

**Speaker**: Bradley Currey, Saint Louis University

**Title**: Heisenberg Frame Sets

**Download Abstract**

**Time**: February 5, 2009. 4:10 pm, room 1312.

**Speaker**: Alexander I. Aptekarev, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow

**Title**: Rational approximants for vector of analytic functions with branch points

**Abstract**: Given a vector of power series expansions at infinity point which allows analytic continuation along any path of complex plane non-intersecting with a finite set of branch points. For this set of functions the Hermite-Pade rational approximants are considered. For the case of one function ? the conjecture of Nuttall (that poles of the diagonal Pade approximants of function with branch points tend to the cut of minimal capacity making the function single-valued) was proven by Stahl. We discuss a generalization for the vector case.

**Time**: January 20, 2009. 4:10 pm, room 1312.

**Speaker**: Andriy Prymak, University of Manitoba

**Title**: Approximation of dilated averages and K-functionals

**Download Abstract**

**Time**: January 13, 2009. 4:10 pm, room 1312.

**Speaker**: Nikos Stylianopoulos, University of Cyprus

**Title**: Fine asymptotics for Bergman orthogonal polynomials over domains with corners

**Download Abstract**

## 2008

**Time**: December 9, 2008. 4:10 pm, room 1312.

**Speaker**: Mike Wakin, Colorado School of Mines

**Title**: Compressive Signal Processing using Manifold Models

**Abstract**: Compressive Sensing (CS) is a framework for signal acquisition built on the premise that a sparse signal can be recovered from a small number of random linear measurements. CS is robust in two important ways: (1) the error in recovering any signal is proportional to its proximity to a sparse signal, and (2) the error in recovering a signal is proportional to the amount of added noise in the measurement vector. In this talk I will describe how a geometric interpretation of CS leads naturally to an extension of CS beyond sparse models to incorporate low-dimensional manifold models for signals. I will discuss how small numbers of random measurements can guarantee a stable embedding of a manifold-modeled signal family in the compressive measurement space, how this leads to analogous robustness guarantees to sparsity-based CS, and how this makes possible new applications in classification, manifold learning, and multi-signal acquisition.

**Time**: December 2, 2008. 4:10 pm, room 1312.

**Speaker**: Truong-Thao Nguyen, City University of New York

**Title**: The tiling phenomenon of 1-bit feedback analog-to-digital converters

**Abstract**: The circuit technology of data acquisition has introduced a high performance technique of analog-to-digital conversion based on the use of coarse quantization compensated by feedback, and called Sigma-Delta modulation. However, while this technique enables data conversion of high resolutions in practice, its design has been mostly developed empirically and its rigorous analysis escapes from standard signal theories. The fundamental difficulty lies in the existence of a nonlinear operation (namely, the quantization) in a recursive process (physically implemented by the feedback). This prevents a tractable and explicit determination of the output in terms of the input of the system. Partial answers to this difficult problem have been recently found as Sigma-Delta modulators have been observed to carry some new interesting mathematical properties. The state vector of the feedback system appears to systematically remain in a *tile* of the state space. This has been the starting point to new research developments involving mathematical tools that are very unusual to the signal processing and communications communities, while simultaneously bringing new results to applied mathematics. This includes ergodic theory, dynamical systems, as well as spectral theory. In this talk, we give an overview on this research, including the mathematical origin of this tiling phenomenon and its consequence to the rigorous signal analysis of Sigma-delta modulators.

**Time**: November 18, 2008. 4:10 pm, room 1312.

**Speaker**: Jeff Hogan, University of Arkansas

**Title**: Clifford analysis and hypercomplex signal processing

**Abstract**: In this talk we attempt to synthesize recent progress made in the mathematical and electrical engineering communities on topics in Clifford analysis and the processing of color images (for example), in particular the construction and application of Clifford-Fourier transforms designed to treat vector-valued signals. Emphasis will be placed on the two-dimensional setting where the appropriate underlying Clifford algebra is the set of quaternions. We’ll conclude with some results and problems in the construction of discrete wavelet bases for color images, and the difficulties encountered in constructing the correct Fourier kernels in dimensions 3 and higher. (This talk is part of the Shanks workshop ‘Nonlinear Models in Sampling Theory’.)

**Time**: November 11, 2008. 4:10 pm, room 1312.

**Speaker**: Simon Foucart, Vanderbilt University

**Title**: A Survey on the Mathematics of Compressed Sensing

**Abstract**: This talk will give an overview of some chosen topics in the theory of Compressed Sensing. Mathematically speaking, one aims at finding sparsest solutions of severely underdetermined linear systems of equations via robust and efficient algorithms. I shall especially discuss the advantages and drawbacks of algorithms based on $\ell_q$-minimization for $0 < q < 1$ compared to the classical $\ell_1$-minimization. This will be based on results — both of positive and negative nature — recently obtained by Chartrand et al., by Gribonval et al., and by Lai and myself.

**Time**: November 4, 2008. 4:10 pm, room 1312.

**Speaker**: Brigitte Forster, Technische Universität München

**Title**: Shift-invariant spaces from rotation-covariant functions

**Abstract**: We consider shift-invariant multiresolution spaces generated by rotation-covariant functions $\rho$ in $\mathbb{R}^2$. To construct corresponding scaling and wavelet functions, $\rho$ has to be localized with an appropriate multiplier, such that the localized version is an element of $L^2(\mathbb{R}^2)$. We consider several classes of multipliers and show a new method to improve regularity and decay properties of the corresponding scaling functions and wavelets. The wavelets are complex-valued functions, which are approximately rotation-covariant and therefore behave as Wirtinger differential operators. Moreover, our class of multipliers gives a novel approach for the construction of polyharmonic B-splines with better polynomial reconstruction properties. The method works not only on classical lattices, such as the cartesian or the quincunx, but also on hexagonal lattices.

**Time**: October 28, 2008. 4:10 pm, room 1312.

**Speaker**: Rick Chartrand, Los Alamos National Laboratory

**Title**: Nonconvex compressive sensing: getting the most from very little information (and the other way around).

**Abstract**: In this talk we’ll look at the exciting, recent results showing that most images and other signals can be reconstructed from much less information than previously thought possible, using simple, efficient algorithms. A consequence has been the explosive growth of the new field known as compressive sensing, so called because the results show how a small number of measurements of a signal can be regarded as tantamount to a compression of that signal. The many potential applications include reducing exposure time in medical imaging, sensing devices that can collect much less data in the first place instead of collecting and then compressing, getting reconstructions from what seems like insufficient data (such as EEG), and very simple compression methods that are effective for streaming data and preserve nonlinear geometry. We’ll see how replacing the convex optimization problem typically used in this field with a nonconvex variant has the effect of reducing still further the number of measurements needed to reconstruct a signal. A very surprising result is that a simple algorithm, designed only for finding one of the many local minima of the optimization problem, typically finds the global minimum. Understanding this is an interesting and challenging theoretical problem. We’ll see examples, and discuss algorithms, theory, and applications.

**Time**: October 14, 2008. 4:10 pm, room 1312.

**Speaker**: Akram Aldroubi, Vanderbilt University

**Title**: Compressive Sampling via Huffman codes.

**Abstract**: Let $x$ be some vector in $\R^n$ with at most $k$ much less than $n$ nonzero components (i.e., $x$ is a sparse vector). We wish to determine $x$ from inner products $\{y_i=a_i\dot x\}_{i=1}^m$, the samples. How can we determine a set of $m$ vectors $\{a_i\}$ such that $x$ can be completely determined from the samples $\{y_i=a_i\dot x\}_{i=1}^m$ by a computationally efficient, stable algorithm. The recent theory of compressed sampling addresses this problem using two main approaches: the geometric approach and the combinatorial approach. In this talk I will present a new information theoretic approach and use results from the theory of Huffman codes to construct a sequence of binary sampling vectors to determine a sparse vector $x$. Unlike the standard approaches, this new method is sequential and adaptive in the sense that each sampling vector depends on the previous sample value. The number of measurements we need is no more than $O(k\log n)$ and the reconstruction is $O(k)$ which is better than any other method.

**Time**: October 7, 2008. 4:10 pm, room 1312.

**Speaker**: Andrii Bondarenko, Kyiv National Taras Shevchenko University

**Title**: New asymptotic estimates for spherical designs.

**Abstract**: The equal weight quadrature formula on the sphere S^n which is exact for all polynomials of n+1 variables and of total degree t is called spherical t-design. We will consider two approaches for constructing good spherical designs for large parameters n and t, which improve essentially the previous upper bounds for minimal number of points in spherical t-design and confirm the well known conjecture of Korevaar and Meyers. We will also show the connection of this area with energy problems, lattices and group theory.

**Time**: September 23, 2008. 4:10 pm, room 1312.

**Speaker**: Akram Aldroubi, Vanderbilt University

**Title**: Invariance of shift-invariance spaces.

**Abstract**: A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. We will characterize those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.

**Time**: September 16, 2008. 4:10 pm, room 1312.

**Speaker**: Hendrik Speleers, Katholieke Universiteit Leuven

**Title**: From PS splines to QHPS splines.

**Abstract**: Powell-Sabin (PS) splines are C^{1}-continuous quadratic macro-elements defined on conforming triangulations. They can be represented in a compact normalized spline basis with a geometrically intuitive interpretation involving control triangles. These triangles can be used to interactively change the shape of a PS spline in a predictable way. In this talk we discuss a hierarchical extension of PS splines, the so-called quasi-hierarchical Powell-Sabin (QHPS) splines. They are defined on a hierarchical triangulation obtained through (local) triadic refinement. For this spline space a compact normalized quasi-hierarchical basis can be constructed. Such a basis retains the advantages of the PS spline basis: the basis functions have a local support, they form a convex partition of unity, and control triangles can be defined. In addition, they admit local subdivision in a very natural way. These properties of QHPS splines are appropriate for local adaptive approximation and modelling.

**Time**: September 9, 2008. 4:10 pm, room 1312.

**Speaker**: Larry Schumaker, Vanderbilt University

**Title**: Dimension of Spline Spaces on T-Meshes.

**Abstract**: A T-mesh $\Delta$ is obtained from a tensor-product mesh by removing certain edges to create a partition with one or more T-nodes. Given $0 \le r_1 \le d_1$ and $0 \le r_2 \le d_2$, we define an associated spline space $S^{r_1,r_2}_{d_1,d_2}(\Delta)$ as the space of functions in $C^{r_1,r_2}$ whose restrictions to the rectangles of the partition are tensor polynomials in $P_{d_1,d_2}$. In this talk we discuss the problem of computing the dimension of these spline spaces. In particular, we give various lower bounds which lead to exact formulae in some cases. We also discuss extensions to more than two variables, and also some results for more general L-meshes. Finally, we conclude with several enticing open questions.

**Time**: April 29, 2008. 4:10 pm, room 1310.

**Speaker**: Maxym Yattselev, INRIA Sophia Antipolis

**Title**: Non-Hermitian Orthogonal Polynomials with Varying Weights on an Arc.

**Abstract**: We consider multipoint Pade approximation of Cauchy transforms of complex measures. We show that if the support of a measure is a smooth Jordan arc and the density of this measure is sufficiently smooth, then the diagonal multipoint Pade approximants associated with interpolation schemes that satisfy special symmetry property with respect to this arc converge locally uniformly to the approximated Cauchy transform. The existence of such interpolation schemes is proved for the case where support is an analytic Jordan arc. The asymptotic behavior of Pade approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials.

**Time**: April 15, 2008. 4:10 pm, room 1310.

**Speaker**: Doug Hardin, Vanderbilt University

**Title**: Discrete minimum energy problems and minimal Epstein zeta functions.

**Abstract**: We consider asymptotic properties (as $N\to \infty$) of `ground state’ configurations of $N$ particles restricted to a $d$-dimensional compact set $A\subset {\bf R}^p$ that minimize the Riesz $s$-energy functional $$ \sum_{i\neqj}\frac{1}{|x_{i}-x_{j}|^{s}} $$ for $s>0$. The first part of this talk will consist of an overview of recent results obtained by the `Vanderbilt minimum energy group’ (aka, the ‘couch potatoes’); in the second half I will present related results and conjectures of Cohn, Elkies and Kumar and to recent results of Sarnak and Strömbergsson concerning minimal zeta functions in dimensions 8 and 24.

**Time**: April 8, 2008. 4:10 pm, room 1310.

**Speaker**: Razvan Teodorescu, Los Alamos National Laboratory.

**Title**: Planar Harmonic Growth with Orthogonal Polynomials.

**Abstract**: This talk will cover recent connections between the theory of orthogonal polynomials with deformed Bargmann kernel and harmonic growth of bounded domains. Singular limits and refined asymptotics will also be discussed.

**Time**: February 26, 2008. 4:10 pm, room 1310.

**Speaker**: Qiang Wu, Duke University.

**Title**: Dimension Reduction in Supervised Learning.

**Abstract**: Dimension reduction in supervised setting aims at inferring the data structure that are most relevant to the prediction of the labels. It can be motivated from either predictive models or descriptive models. Starting from a predictive model, we showed the gradient outer product matrix contains the information of relevant features and predictive dimensions. Several well known feature selection and dimension reduction methods follow this criterion either implicitly or explicitly. We designed an algorithm of learning gradients specifically for the small sample size setting using kernel regularization. The asymptotic analysis shows the convergence depends only on the intrinsic dimension of the data and can be fast if the underlying data concentrate on a low dimensional manifold. The gradient estimate was successfully applied to feature selection, dimension reduction, estimation of conditional dependency and task similarity in high dimensional data analysis. Sliced inverse regression (SIR) is a well known and widely used dimension reduction methods in statistics community. It is motivated from a descriptive model. We studied the relation between the gradient out product matrix and covariance matrix of the inverse regression function and found they are locally equivalent in certain sense. This observation not only helps clarify the theoretical comparison between these two methods but also motivates a new algorithm. We developed localized sliced inverse regression (LSIR) for dimension reduction which overcomes the degeneracy problem of original SIR and has the advantage of finding clustering structure in classification problems.

**Time**: February 19, 2008. 4:10 pm, room 1310.

**Speaker**: Abey Lopez, Vanderbilt University.

**Title**: Asymptotic Behavior of Greedy Energy Configurations.

**Abstract**: In this talk we will discuss some results about the asymptotic behavior of certain point configurations called Greedy Energy (GE) points. These points form a sequence which is generated by means of a greedy algorithm, which is an energy minimizing construction. The notion of energy that we consider comes from the Riesz potentials V=1/r^{s} in R^{p}, where s>0 and r denotes the Euclidian distance. It turns out that for certain values of the parameter s, these configurations behave asymptotically like Minimal Energy (ME) configurations. This property will also be discussed in more abstract contexts like locally compact Hausdorff spaces. For other values of s, GE and ME configurations exhibit different asymptotic properties, for example for s>1 on the unit circle. We will discuss other questions like second order asymptotics on the unit circle and weighted Riesz potentials on unit spheres.

**Time**: February 12, 2008. 4:10 pm, room 1310.

**Speaker**: Justin Romberg, Georgia Tech.

**Title**: Compressed Sensing for Next-Generation Signal Acquisition.

**Abstract**: From decades of research in signal processing, we have learned that having a good signal representation can be key for tasks such as compression, denoising, and restoration. The new theory of Compressed Sensing (CS) shows us how a good representation can fundamentally aid us in the acquisition (or sampling) process as well. In this talk will outline the main theoretical results in CS and discuss how the ideas can be applied in next-generation acquisition devices. The CS paradigm can be summarized neatly: the number of measurements (e.g., samples) needed to acquire a signal or image depends more on its inherent information content than on the desired resolution (e.g., number of pixels). The CS theory typically requires a novel measurement scheme that generalizes the conventional signal acquisition process: instead of making direct observations of the signal, for example, an acquisition device encodes it as a series of random linear projections. The theory of CS, while still in its developing stages, is far- reaching and draws on subjects as varied as sampling theory, convex optimization, source and channel coding, statistical estimation, uncertainty principles, and harmonic analysis. The applications of CS range from the familiar (imaging in medicine and radar, high-speed analog-to-digital conversion, and super-resolution) to truly novel image acquisition and encoding techniques.

## 2007

**Time**: December 5, 2007. 4:10 pm, room 1312.

**Speaker**: Tom Lyche, University of Oslo.

**Title**: New Formulas for Divided Differences and Partitions of a Convex Polygon.

**Abstract**: Divided differences are a basic tool in approximation theory and numerical analysis: they play an important role in interpolation and approximation by polynomials and in spline theory. So it is worthwhile to look for identities that are analogous to identities for derivatives. An example is the Leibniz rule for differentiating products of functions. This rule was generalized to divided differences by Popoviciu and Steffensen 70 years ago. To our surprise it was discovered that there were no analog of a 150 year old formula for differentiating composite functions (Faa di Bruno’s formula) and for differentiating the inverse of a function. In this talk I will discuss chain rules and inverse rules for divided differences. The inverse rule turns out to have a surprising and beautiful structure: it is a sum over partitions of a convex polygon into smaller polygons using only nonintersecting diagonals. This provides a new way of enumerating all partitions of a convex polygon with a specified number of triangles, quadrilaterals, and so on. The talk is based on joint work with Michael Floater.f new infinite product representations for trigonometric and hyperbolic functions that have not been known before.

**Time**: November 27, 2007. 4:10 pm, room 1310.

**Speaker**: Yu. A. Melnikov, Middle Tennessee State University.

**Title**: An innovative approach to the derivation of infinite product representations of elementary functions.

**Abstract**: We will report on a curious outcome from the classical method for the construction of Green’s functions for Laplace equation. An innovative technique is developed for obtaining infinite product representations of elementary functions. Some standard boundary value problems are considered posed for two-dimensional Laplace equation on regions of regular configuration. Classical analytic forms of Green’s functions for such problems are compared against those obtained by the method of images. This yields a number of new infinite product representations for trigonometric and hyperbolic functions that have not been known before.

**Time**: November 13, 2007. 4:10 pm, room 1310.

**Speaker**: Minh N. Do, University of Illinois at Urbana-Champaign.

**Title**: Sampling Signals from a Union of Subspaces.

**Abstract**: One of the fundamental assumptions in traditional sampling theorems is that the signals to be sampled come from a single vector space (e.g. bandlimited functions). However, in many cases of practical interest the sampled signals actually live in a union of subspaces. Examples include piecewise polynomials, sparse approximations, nonuniform splines, signals with unknown spectral support, overlapping echoes with unknown delay and amplitude, and so on. For these signals, traditional sampling schemes are either inapplicable or highly inefficient. In this paper, we study a general sampling framework where sampled signals come from a known union of subspaces and the sampling operator is linear. Geometrically, the sampling operator can be viewed as projecting sampled signals into a lower dimensional space, while still preserves all the information. We derive necessary and sufficient conditions for invertible and stable sampling operators in this framework and show that these conditions are applicable in many cases. Furthermore, we find the minimum sampling requirements for several classes of signals, which indicates the power of the framework. The results in this paper can serve as a guideline for designing new algorithms for many applications in signal processing and inverse problems.

**Time**: October 16, 2007. 4:10 pm, room 1310.

**Speaker**: Kourosh Zarringhalam, Vanderbilt University.

**Title**: Chaotic Unstable Periodic Orbits, Theory and Applications.

**Abstract**: We will present a control scheme for stabilizing the unstable periodic orbits of chaotic systems and investigate the properties of these orbits. These approximated chaotic unstable periodic orbits are called cupolets (Chaotic Unstable Periodic Orbit-lets). The cupolet transformation can be regarded as an alternative to Fourier and wavelet transformations and can be used in variety of applications such as data and music compression, as well as image and video processing. We will also investigate the shadowability of cupolets and present a shadowing theorem, suitable for computational purposes, that provides a way to establish the existence of true periodic and non-periodic orbits near the approximated ones.

**Time**: October 9, 2007. 4:10 pm, room 1310.

**Speaker**: Simon Foucart, Vanderbilt University.

**Title**: Condition numbers of finite-dimensional frames.

**Abstract**: First, motivated by some problems in spline theory, we will introduce the notion of condition number of a basis. We will then review some results on best conditioned bases, and examine how they relate to minimal projections. Finally, the notion of condition number will be extended — in finite dimension — to frames. This work is in progress and highlights some intriguing questions in connection with the geometry of Banach spaces.

**Time**: October 2, 2007. 4:10 pm, room 1310.

**Speaker**: Carolina Beccari, University of Bologna.

**Title**: Tension-controlled interpolatory subdivision.

**Abstract**: Subdivision generates a smooth curve/surface as the limit of a sequence of successive refinements applied to an initial polyline/mesh. Although subdivision curves and surfaces can be generated either through interpolation or approximation of the initial control net, interpolatory refinements have been traditionally considered less attractive than approximatory methods, due to the poor visual quality of their limit shapes. This problem will be addressed taking into account the novel notions of non-stationarity and non-uniformity in order to include in subdivision models the important capability of tension control together with the capacity of reproducing prescribed curves and conic sections, that is peculiar to the NURBS representation. To this aim we will explore the definition of subdivision schemes featured by the presence of tension parameters associated with the edges in the initial control polygon/net.Since these parameters give us the possibility of locally adjusting the shape of the limit curve, they can be used both to produce a nice-looking interpolation of the initial control points and to achieve the exact modeling of circular arcs, surfaces of revolution and quadrics.

**Time**: September 25, 2007. 3:10 pm, room 1310.

**Speaker**: Rene Vidal, Johns Hopkins University.

**Title**: Generalized Principal Components Analysis.

**Abstract**: Over the past two decades, we have seen tremendous advances on the simultaneous segmentation and estimation of a collection of models from sample data points, without knowing which points correspond to which model. Most existing segmentation methods treat this problem as “chicken-and-egg”, and iterate between model estimation and data segmentation. This lecture will show that for a wide variety of data segmentation problems (e.g. mixtures of subspaces), the “chicken-and-egg” dilemma can be tackled using an algebraic geometric technique called Generalized Principal Component Analysis (GPCA). This technique is a natural extension of classical PCA from one to multiple subspaces. The lecture will touch upon a few motivating applications of GPCA in computer vision, such as image/video segmentation, 3-D motion segmentation or dynamic texture segmentation, but will mainly emphasize the basic theory and algorithmic aspects of GPCA.

**Time**: September 18, 2007. 4:10 pm, room 1310.

**Speaker**: Romain Tessera, Vanderbilt University.

**Title**: Finding left inverses for a class of operators on l^p(Z^d) with concentrated support.

**Abstract**: We will expose various generalizations of the following recent theorem (due to Aldroubi, Baskarov, Krishtal): Let A=(a_{x,y}) be a matrix indexed by Z^d x Z^d such that a_{x,y}=0 whenever |x-y|>m for some m. Assume that A has bounded coefficients and is bounded below as an operator on l^p for some p in [1,infty]. Then it has a left-inverse B which is bounded on l^q for all q in [1,infty]. The proof that we propose is quite different from the one of Aldroubi, Baskarov, Krishtal. It essentially relies on a basic geometric property of Z^d, and hence works in a more general setting.

**Time**: September 11, 2007. 4:10 pm, room 1310.

**Speaker**: Larry Schumaker, Vanderbilt University.

**Title**: Computing Bivariate Splines in Scattered Data Fitting and the FEM Method.

**Abstract**: A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at the same time. Typically this involves solving a (possible large) system of linear equations. Examples include several well-known methods for fitting scattered data, such as the minimal energy, least-squares, and penalized least-squares methods. Finite-element methods for solving boundary-value problems are also of this type. We show how these types of globally-defined splines can be efficiently computed, provided we work with spline spaces with stable local bases.

**Time**: April 19, 2007. 2:10 pm, room 1310.

**Speaker**: Laurent Baratchart, INRIA, Sophia Antipolis.

**Title**: Dirichlet problems and Hardy spaces for the real Beltrami equation.

**Abstract**: Motivated by extremal problems connected with locating the plasma boundary in a Tokamak vessel, we consider Dirichlet problems for the real Beltrami equation: \partial f/\partial{\bar z}=\nu\overline{\partial f/\partial z} on the disk or the annulus. We show the existence of a unique solution with given real part in certain Sobolev spaces of the boundary for bounded measurable nu bounded away from below, the density of traces of solutions on subarcs of the boundary, and the existence of solutions in Hardy-type classes defined through the finiteness of L^p means on inner circles. We briefly discuss the analog of classical extremal problems in this context.

**Time**: April 17, 2007. 4:10 pm, room 1312.

**Speaker**: Casey Leonetti, Vanderbilt University.

**Title**: Error Analysis of Frame Reconstruction from Noisy Samples

**Abstract**: This talk addresses the problem of reconstructing a continuous function from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean zero and variance sigma-squared. We sample the continuous function f on the uniform lattice (1/m)Z^d, and show for large enough m that the variance of the error between the frame reconstruction from noisy samples of f and the function f evaluated at each point x behaves like sigma-squared divided by m^d times a (best) constant C_x. We also prove a similar result in the case that our data are weighted-average samples of f corrupted by additive noise. Joint work with Akram Aldroubi and Qiyu Sun.

**Time**: April 11, 2007. 4:10 pm, room 1312.

**Speaker**: Ju-Yi Yen, Vanderbilt University.

**Title**: Multivariate Jump Processes in Financial Returns.

**Abstract**: We apply a signal processing technique known as independent component analysis (ICA) to multivariate financial time series. The main idea of ICA is to decompose the observed time series into statistically independent components (ICs). We further assume that the ICs follow the variance gamma (VG) process. The VG process is evaluated by Brownian motion with drift at a random time given by a gamma process. We build a multivariate VG portfolio model and analyze empirical results of the investment.

**Time**: April 4, 2007. 4:10 pm, room 1312.

**Speaker**: Kasso Okoudjou, University of Maryland.

**Title**: Uncertainty principle for fractals, graphs, and metric measure spaces.

**Abstract**: We formulate and prove weak uncertainty principles for functions defined on fractals, graphs and more generally on metric measure spaces. In particular, this uncertainty inequality is proved under different assumptions such as an appropriate measure growth condition with respect to a specific metric, or in the absence of such a metric, we assume the Poincare inequality and the reverse volume doubling property.

**Time**: March 21, 2007. 4:10 pm, room 1312.

**Speaker**: Johann S. Brauchart, Vanderbilt University.

**Title**: Optimal logarithmic energy points on the unit sphere in $\mathbb{R}^{d+1}$, $d\geq2$.

**Abstract**: We study minimum energy point charges on the unit sphere in $\Rset^{d+1}$, $d\geq2$, that interact according to the logarithmic potential $\log(1/r)$, where $r$ is the Euclidean distance between points. Such optimal $N$-point configurations are uniformly distributed as $N\to\infty$. We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate is of order $\mathcal{O}(N^{-1/(d+2)})$. Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term $(1/d)(\log N)/N$ in the asymptotical expansion of the optimal energy. Previously, the latter has been known for the unit sphere in $\mathbb{R}^{3}$ only. From the proof of our discrepancy estimates we get an upper bound for the error of integration for polynomials of degree at most $n$ when using an equally-weighted numerical integration rule $\numint_{N}$ with the $N$ nodes forming an optimal logarithmic energy configuration. This bound is $C_{d} ( N^{1/d} / n )^{-d/2} \| p \|_{\infty}$ as $n/N^{1/d}\to0$.

**Time**: March 14, 2007. 4:10 pm, room 1312.

**Speaker**: Elena Berdysheva, University of Hohenheim, Germany.

**Title**: On Tur\’an’s Problem for $\ell$-1 Radial, Positive Definite Functions.

**Abstract**: Tur\’an’s problem is to determine the greatest possible value of the integral $\int_{{\mathbb R}^d}f(x)\,dx / f(0)$ for positive definite functions $f(x)$, $x \in {\mathbb R}^d$, supported in a given convex centrally symmetric body $D \subset {\mathbb R}^d$. In this talk we consider the Tur\’an problem for positive definite functions of the form $f(x) = \varphi(\|x\|_1)$, $x \in {\mathbb R}^d$, with $\varphi$ supported in $[0,\pi]$. An essential part of the talk is devoted to the planar case ($d=2$), in this case we could settle and solve the corresponding discrete problem. Some of our results are proved for an arbitrary dimension. Joint work with H. Berens (University of Erlangen-Nuremberg, Germany).

**Time**: February 14, 2007. 4:10 pm, room 1310.

**Speaker**: Ming-Jun Lai, University of Georgia.

**Title**: Bivariate Splines for Statistical Applications.

**Abstract**: I will use bivariate splines for functional data analysis and rank restricted approximation of data.

**Time**: February 7, 2007. 4:10 pm, room 1312.

**Speaker**: Maxim Yattselev, Vanderbilt University.

**Title**: On uniform convergence of AAK approximants.

**Abstract**: In this talk we present some results on uniform convergence of AAK approximants to functions of the form $$F(z) = \int_{[a,b]}\frac{1}{z-t}\frac{s_{\alpha,\beta}(t)s(t)dt}{\sqrt{(t-a)(b-t)}}+R(z), \;\;\; \alpha,\beta\in[0,1/2),$$ where $s_{\alpha,\beta}(t)=(t-a)^\alpha(b-t)^\beta$, $R$ is a rational function analytic at infinity having no poles on $[a,b]$, and $s$ is a complex-valued Dini continuous nonvanishing function on $[a,b]$ with an argument of bounded variation there.

**Time**: January 31, 2007. 4:10 pm, room 1312.

**Speaker**: Alexander Aptekarev, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences.

**Title**: Discrete Entropy of Orthogonal Polynomials.

**Abstract**: Information entropy has been introduced by Shanon as a density functional for measuring of uncertainness of distributions. In quantum mechanics this functional is used to provide more sharp bounds in uncertainness relations (sharper than Heisenberg uncertainness relation for the first moments – i.e. for the mathematical expectations). Since the density of the distributions of many classical quantum mechanical systems (oscillators, Coulomb potential, hydrogen-like atoms) are represented by means of orthogonal polynomials, there is a demand from quantum physicists to compute entropy of orthogonal polynomials. In this talk we present some computational and explicit results.

**Time**: January 24, 2007. 4:10 pm, room 1312.

**Speaker**: Alex Powell, Vanderbilt University.

**Title**: Finding good dual frames for reconstructing quantized frame expansions.

**Abstract**: This talk will begin by reviewing the basics of Sigma-Delta quantization. Sigma-Delta quantization is an algorithm for digitizing/rounding the coefficients in a redundant signal expansion. We shall work in the setting of finite frames and address the problem of finding dual frames which are better suited for signal reconstruction than the canonical dual frame.

## 2006

**Time**: December 5, 2006. 4:00 pm, room 1310.

**Speaker**: Peter Grabner, Graz University of Technology.

**Title**: Periodicity Phenomena in the Analysis of Algorithms and Related Dirichlet Series.

**Abstract**: Average case analysis of algorithms studies the behaviour of an algorithm under a probabilistic model on the data. Many algorithms have a recursive structure, which gives a recursion for the average performance. In many cases, the asymptotic behaviour of the solutions of this recursion shows a periodicity in the logarithmic scale, which corresponds to complex poles of the generating Dirichlet series. We discuss a method for acceleration of convergence of such series and give several examples for its application.

**Time**: November 28, 2006. 3:00 pm, room 1310.

**Speaker**: Nikos Stylianopoulos, University of Cyprus.

**Title**: Finite-term recurrence relations for planar orthogonal polynomials.

**Abstract**: We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. (A report of joint work with Mihai Putinar.)

**Time**: November 14, 2006. 4:00 pm, room 1310.

**Speaker**: Yuan Xu, University of Oregon.

**Title**: Radon transforms, orthogonal polynomials and CT.

**Abstract**: The central problem for computered tomography (CT) is to reconstruct a function (an image) from a finite set of its Radon projections. We propose a reconstruction algorithm, called OPED, based on Orthogonal Polynomial Expansion on the Disk. The algorithm works naturally with the fan data and can be implemented efficiently. Furthermore, it is proved that the algorithm converges uniformly under a mild condition on the function. Numerical experiments have shown that the method is fast, stable, and has a small global error.

**Time**: Novmeber 7, 2006. 4:00 pm, room 1310.

**Speaker**: Darrin Speegle, St. Louis University.

**Title**: The Feichtinger Conjecture for special classes of frames.

**Abstract**: Feichtinger conjectured that every frame for a Hilbert space can be partitioned into the finite union of sets, each of which is a Riesz basis for its closed linear span. It was quickly realized that this conjecture was closely related to the paving problem for matrices, and thus to the Kadison-Singer problem. More recently, it has been shown that settling the Feichtinger Conjecture is equivalent to solving the paving problem. In this talk I will review the partial results on the paving problem, primarily by Bourgain and Tzafriri, and translate them into partial results on the Feichtinger Conjecture. Then, I will describe the progress that has been made for Gabor frames, wavelet frames and frames of exponentials. For these restricted classes of frames, it is not clear whether settling the Feichtinger Conjecture is equivalent to solving the corresponding paving problems. Despite progress, the Feichtinger Conjecture remains open even in this restricted setting.

**Time**: October 10, 2006. 4:00 pm, room 1310.

**Speaker**: Bruce Atkinson, Samford University.

**Title**: An introduction to Markovian image models.

**Abstract**: A random field is a probability measure on the set of images, where an image is an assignment of grey levels to vertices of a graph. We use the Gibbs sampler to realize a field, and explain how the sampler is improved if the field is Markovian. We assume a given image is a realization of a Markovian field and the observed image is a local degradation of it. The posterior distribution of the true image, given the degraded one, is also Markovian and a modification of the Gibbs sampler (an analog of simulated annealing) is used to restore the true image as a maximum likelihood estimate based on the posterior distribution.

**Time**: October 3, 2006. 4:00 pm, room 1310.

**Speaker**: Doug Hardin, Vanderbilt University.

**Title**: Orthogonal wavelets centered on non-uniform knot sequences.

**Abstract**:We develop a general notion of orthogonal non-uniform wavelets centered on a knot sequence. As an application, we construct C^0 and C^1 piecewise polynomial multiwavelets for a knot sequence associated with a golden-mean refinement scheme.

**Time**: September 26, 2006. 4:00 pm, room 1310.

**Speaker**: Larry Schumaker, Vanderbilt University.

**Title**: Bounds on the dimension of trivariate spline spaces.

**Abstract**:We discuss recent results with Peter Alfeld giving upper and lower bounds on the dimensions of trivariate spline spaces defined on tetrahedral partitions. The results hold for general partitions and for all degrees of smoothness r and polynomial degrees d.

**Time**: September 19, 2006. 4:00 pm, room 1310.

**Speaker**: Simon Foucart, Vanderbilt University.

**Title**: The Orthogonal Projector Onto Splines — Ongoing Development.

**Abstract**:A few years ago, the long-standing conjecture that the max-norm of the orthogonal spline projector is bounded independently of the underlying knot sequence was settled. However, a delicate question remains open, namely: what is the exact value [or order] of the bound? I will present some precise estimates for splines of low smoothness. I will also discuss some approaches for answering the previous question.

**Time**: September 12, 2006. 4:00 pm, room 1310.

**Speaker**: Fumiko Futamura, Vanderbilt University

**Title**: Localized Operators and the Construction of Localized Frames.

**Abstract**: A frame for a Hilbert space is a kind of generalized orthonormal basis which is useful in signal processing. A localized frame is a frame whose elements are “well-localized”, in the sense that the inner products of their elements decay as the differences of their indices increase. Grochenig in 2004 proved that localized frames for Hilbert spaces extend to frames for a family of associated Banach spaces. We generalize localized frames to the operator setting, and say an operator is localized with respect to given frames if there is an off-diagonal decay of the matrix representation of an operator with respect to the frames. We prove that operators localized with respect to localized frames are bounded on the same family of Banach spaces, and that they can be used in the construction of new localized frames. We also consider the special case where the frames are unitary shifts of a single atom function.

**Time**: September 5, 2006. 4:00 pm, room 1310.

**Speaker**: Mike Neamtu, Vanderbilt University

**Title**: Splines on Triangulations for CAGD.

**Abstract**: In this talk I will discuss the question of whether piecewise (algebraic) polynomials are the appropriate tools for defining splines in CAGD.

**Time**: April 29, 2006. 4:10-5 pm, room 1431.

**Speaker**: Ed Saff, Vanderbilt University

**Title**: Asymptotics for Polynomial Zeros: Beware of Predictions from Plots.

**Abstract**:

**Time**: April 20, 2006. 4:10-5 pm, room 1308.

**Speaker**: David Benko (Western Kentucky University).

**Title**: Approximation by homogeneous polynomials.

**Abstract**: Let K be a convex origin symmetric surface in R^d. Kroo conjectures that any continuous function on K can be uniformly approximated by a sum of two homogeneous polynomials. Using potential theory and weighted polynomials we resolve this problem on the plane. We also give a positive answer in higher dimensions under a smoothness condition on K.

**Time**: April 11, 2006. 4:10-5 pm, room 1308.

**Speaker**: Vasily Prokhorov (Univ. South Alabama and Vanderbilt).

**Title**: On Estimates for the Ratio of Errors in Best Rational Approximation of Analytic Functions.

**Abstract**: Let E be an arbitrary compact subset of the extended complex plane with non-empty interior. For a function f continuous on E and analytic in the interior of E denote by $\rho_n(f; E)$ the least uniform deviation of f on E from the class of all rational functions of order at most n. We will show that if K is an arbitrary compact subset of the interior of E, then $ \prod_{k=0}^n (\rho_k(f; K) /\rho_k(f; E) ),$ the ratio of the errors in best rational approximation, converges to zero geometrically as $n \to \infty$ and the rate of convergence is determined by the capacity of the condenser (\partial E, K).

**Time**: April 4, 2006. 4:10-5 pm, room 1308.

**Speaker**: Arthur David Snider, University of South Florida.

**Title**: High Dynamic Range Resampling for Software Radio.

**Abstract**:The classic problem of recovering arbitrary values of a band-limited signal from its samples has an added compli- cation in software radio applications; namely, the resampling calculations inevitably fold aliases of the analog signal back into the original bandwidth. The phenomenon is quantifified by the spur-free dynamic range. We demonstrate how a novel application of the Remez (Parks-McClellan) algorithm permits optimal signal recovery and SFDR, far surpassing state-of-the-art resamplers.

**Time**: March 28,2006. 4:10-5 pm, room 1308.

**Speaker**: Maxim Yattselev, Vanderbilt University.

**Title**: Strong asymptotics on a segment and its application to meromorphic and Pad\’e approximation (joint work with Prof. L. Baratchart, INRIA, Sophia Antipolis, France)

**Abstract**:We consider a strong (Szeg\H{o}-type) asymptotics for polynomials orthogonal with varying complex measures on a segment. We take the approach of G. Baxter of transferring the problem to the unit circle and dealing with the symmetric rational functions. We apply this result to obtain the uniform convergence and the distribution of poles of meromorphic and Pad\’e approximants of complex Cauchy transforms.

**Time**: March 20,2006. 4:10-5 pm, room 1431.

**Speaker**: Laurent Baratchart (INRIA).

**Title**: Bounded Extremal Problems in Hardy Spaces of the ball in $ {\bf R}^n$.

**Abstract**:Carleman-type integral formulas for the asymptotic recovery of holomorphic functions in the disk from partial boundary data turn out to solve extremal problems where a function given on a subset of the circle is to be best-approximated in the $L2$-norm on that subset by a $H2$- function subject to certain constraints on the rest of the circle. We develop the case of a $L2$ constraint and of a pointwise constraint. The approximant can be further characterized as the solution to a spectral Toeplitz equation, and this formulation carries over to Stein-Weiss divergence free Hardy spaces of the ball in ${\bf R}^n$ where it solves a similar approximation problem on the sphere (the case of a half-space is also covered this way via the Kelvin transform). The extremal problem can itself be viewed as a regularization scheme for inverse Dirichlet-Neumann problems.

**Time**: February 13, 2006. 4:10-5 pm, room 1431.

**Speaker**: Ozgur Yilmaz (University of British Columbia).

**Title**: The Role of Sparsity in Blind Source Separation. (Shanks Workshop).

**Abstract**: Certain inverse problems can be solved quite efficiently if the solution is known to have a sparse atomic decomposition with respect to some basis or frame in a Hilbert space. One particular example of such an inverse problem is the so-called cocktail party (or blind source separation) problem: Suppose we use a few microphones to record several people speaking simultaneously. How can we separate individual speech signals from these mixtures? In this talk, I will describe an algorithm adressing the blind source separation problem when the number of speakers is larger than the number of available mixtures. The algorithm is based on the key observation that Gabor expansions of speech signals are sparse. The separation is done in two stages: First, the “mixing matrix” A is estimated via clustering. Next, the Gabor coefficients of individual sources are computed by solving many q-norm minimization problems of type {min ||x||_q subject to Ax=b}. Several choices for the value of q will be compared.

**Time**: February 7, 2006. 4:10-5 pm, room 1308.

**Speaker**: Yuliya Babenko, Vanderbilt University.

**Title**: On asymptotically optimal partitions and the error of approximation by linear and bilinear splines.

**Abstract**: In this talk we shall present exact asymptotics of the optimal error of linear spline interpolation of an arbitrary function in various settings, in particular for the case of $L_p$-norm, $1\leq p \leq \infty$, and $f \in C^2([0,1]^2)$, and for the case of $L_{\infty}$-norm and $f \in C^2([0,1]^d)$. We shall present review of existing results as well as a series of new ones. Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations for linear interpolation. Similar results are obtained for near interpolating bilinear splines.

**Time**: January 31, 2006. 4-10-5pm, room 1431.

**Speaker**: Maxym Yattselev, Vanderbilt University.

**Title**: Meromorphic Approximants for Complex Cauchy Transforms with Polar Singularities.

**Abstract**: We consider a distribution of poles and convergence of meromorphic approximants to functions of the type $$\int\frac{d\mes(t)}{z-t}+R(z),$$ where $R$ is a rational function vanishing at infinity and $\mu$ is a complex measure with the regular support on $(-1,1)$ and whose argument is of bounded variation.

## 2005

**Time**: December 6, 2005. 4:10-5 pm, room 1431.

**Speaker**: Casey Leonetti, Vanderbilt University.

**Title**: Non-Uniform Sampling and Reconstruction From Sampling Sets with Unknown Jitter.

**Abstract**: This talk will address the problem of non-uniform sampling and reconstruction in the presence of jitter. In sampling applications, the countable set X on which a signal f is sampled is not precisely known. Two main questions are considered. First, if sampling a function f on the countable set X leads to unique and stable reconstruction of f, then when does sampling on the set X’, a perturbation of X, also lead to unique and stable reconstruction? Second, if we attempt to recover a sampled function f using the reconstruction operator corresponding to the sampling set X (because the precise sample points are unknown), is the recovered function a good approximation of the original f? Based on work with Akram Aldroubi.

**Time**: November 29, 2005. 4:10-5 pm, room 1431.

**Speaker**: Vincent Lunot, INRIA, France.

**Title**: A Zolotarev Problem with Application to Microwave Filters.

**Abstract**:

**Time**: November 15,2005. 4:10-5 pm, room 1431.

**Speaker**: Dr. Karin Hunter, University of Stellenbosch, South Africa.

**Title**: A class of symmetric interpolatory subdivision schemes.

**Abstract**: The well known Dubuc-Deslauriers subdivision masks are symmetric, interpolatory and satisfy a certain polynomial filling property. Here we define a class of symmetric interpolatory masks that include the Dubuc-Deslauriers masks and then give a method to generate masks in this class. We conclude by providing a condition for convergence of a subdivision scheme for a subset of masks in this class.

**Time**: November 8, 2005. 4:10-5 pm, room 1431.

**Speaker**: Jorge Stolfi, Institute of Computing, State University of Campinas (Brazil).

**Title**: Splines on the Sphere (A View from the Other Hemisphere).

**Abstract**: Polynomial splines on the sphere with triangular topology were defined and thoroughly studied by Alfeld, Neamtu and Schumaker ca. 1996. In this talk we will review the theory of spherical polynomials, their relation to spherical harmonics, and the basics of spherical polynomial spliines. We will then discuss the use of such splines for function approximation and the integration of differential equations on the sphere. (Joint work with Anamaria Gomide)

**Time**: November 1, 2005. 4:10-5 pm, room 1431.

**Speaker**: Alex Powell, Vanderbilt University.

**Title**: Analog to digital conversion for finite frame expansions.

**Abstract**: We shall dicuss the mathematical aspects of analog-to-digital conversion for redundant signal expansions. We restrict ourselves to the case of finite dimensional data, and consider the naturally associated class of signal expansions given by finite frames. Our focus will be on a special class of algorithms, known as Sigma-Delta quantizers, which are related to error diffusion. We explain the basics of Sigma-Delta schemes and point to ongoing directions of research such as error estimates and stability theorems.

**Time**: October 18, 2005. 4:10-5 pm, room 1431.

**Speaker**: Prof. Terry P. Lybrand, Vanderbilt University Center for Structural Biology.

**Title**: Computer simulation of biomacromolecules and complexes.

**Abstract**: Computational approaches have become indispensable for study of large biological molecules over the past twenty-plus years. It is also possible, at least in principle, to use simulations and other computational techniques to predict structural and thermodynamic properties. In my group, we are interested primarily in equilibrium thermodynamic properties of biomolecules and complexes, so we use statistical mechanical calculations to estimate these properties. Direct calculation of a partition function for these complex systems is not possible, so we utilize simulation methods like molecular dynamics or (less frequently) Monte Carlo to calculate approximate partition functions via ensemble averaging. I will present some general details of our calculations, discuss common problems and limitations we encounter, and highlight some areas where we hopefully can take advantage of recent mathematical developments to improve our calculations.

**Time**: September 27, 2005. 4:10-5 pm, room 1431.

**Speaker**: Yuliya Babenko, Vanderbilt University.

**Title**: On asymptotically optimal methods of approximation by linear and bilinear splines.

**Abstract**: In this talk we shall present exact asymptotics of the optimal error in different metrics of linear and bilinear spline interpolation of an arbitrary function $f \in C^2([0,1]^2)$. We shall present review of existing results as well as a series of new ones. Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations (in the case of interpolation by linear splines) and non uniform rectangular partitions (in the case of interpolation by bilinear splines).

**Time**: September 20, 2005. 4:10-5pm, room 1431.

**Speaker**: Larry Schumaker, Vanderbilt University.

**Title**: Trivariate $C^r$ Polynomial Macro-Elements.

**Abstract**: $C^r$ macro-elements defined in terms of polynomials of degree $8r+1$ on tetrahedra are analyzed. For $r=1,2$, these spaces reduce to well-known macro-element spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.

**Time**: September 13, 2005. 4:10-5pm, room 1431.

**Speaker**: Kerstin Hesse, Vanderbilt University.

**Title**: Optimal Cubature on the Sphere.

**Abstract**: In this talk I will present results from joint work with Ian H.\,Sloan on cubature (or numerical integration) on the unit sphere $S^2$ in Sobolev spaces. We prove that the worst-case error $e(H^s;Q_m)$ of an $m$-point cubature rule $Q_m$ in the Sobolev space $H^s=H^s(S^2)$, $s>1$, has the optimal order $O(m^{-s/2})$. To achieve this we need two results: On the one hand, we show that for any $m$-point cubature rule $Q_m$ the worst-case cubature error satisfies $e(H^s;Q_m)\geq C\,m^{-s/2}$, with a constant $C$ independent of the rule $Q_m$ (lower bound). On the other hand, we derive an upper bound for the optimal order of the worst-case error by identifying an infinite sequence $(Q_m)$ of $m$-point cubature rules (where $m$ is from an infinite set of natural numbers) for which the worst-case cubature error has an upper bound of the order $O(m^{-s/2})$. The results extend in a natural way to the Sobolev spaces $H^s(S^d)$, where $s>d/2$, on spheres $S^d$ of arbitrary dimension $d>2$ (proof of the lower bound by myself and proof of the upper bound jointly with Johann S.\,Brauchart).

**Time**: April 19, 2005. 4:10-5 pm, room 1206.

**Speaker**: Doron Lubinsky, Georgia Tech.

**Title**: Which weights on R admit Jackson theorems?

**Abstract**: Let W : R ! (0;1) be continuous. Does W admit a Jackson or Jackson-Favard Inequality? That is, does there exist a sequence f´ng1 n=1 of positive numbers with limit 0 such that for 1 · p · 1; inf deg(P)·n k (f ¡ P)W kLp(R)· ´n k f0W kLp(R) for all absolutely continuous f with k f 0W kLp(R) ¯nite? We show that such a theorem is true i® both lim x!1 W (x) Z x 0 W¡1 = 0 and lim x!1Ãsup [0;x] W¡1!Z 1 x W = 0; with analogous limits as x ! ¡1. In particular W (x) = exp (¡jxj) does not admit a Jackson theorem, although it is well known that W (x) = exp (¡jxj®) ; ® > 1, does. We also construct weights that admit an L1 but not an L1 Jackson theorem (or conversely). The talk will be introductory, and might be accessible to those to whom Jackson and Bernstein sound like the directors of a large corporation.

**Time**: April 5, 2005. 4:10-5 pm, room 1431.

**Speaker**: Hong-Tae Shim, Visiting Professor, Sun Moon University, South Korea.

**Title**: On Gibbs phenomenon in wavelet expansions: its history and development.

**Abstract**: When a function with jump discontinuity is represented by the trigonometric series, one can observe that its graph exhibits overshoot or downshot near the point of discontinuity. This phenomenon is called the Gibbs’ phenomenon, which has been recognized for over a century. However, Gibbs phenomenon is not the special quirk of trigonometric series. It has been shown to exist for many natural approximation, e.g., those involving Fourier series and other classical orthogonal expansions. In this talk, brief history and illustrations are given. We mainly focus on Gibbs phenomenon in wavelet expansions and provide a way to go around it.

**Time**: March 29, 2005. 4:10-5 pm, room 1431.

**Speaker**: Gitta Kutyniok, Univ. Giessen, Germany.

**Title**: Density of irregular wavelet systems.

**Abstract**: Density conditions have recently turned out to be a useful and elegant tool for studying irregular wavelet systems. In this talk we will discuss necessary and sufficient density conditions on the set of parameters for an irregular wavelet system to constitute a frame. In particular, we will derive a necessary condition on the relationship between the affine density, the frame bounds, and the admissibility condition. Several implications of this relationship will be studied. Moreover, we will prove that density conditions can also be used to characterize existence of wavelet frames, thus serving in particular as sufficient conditions.

**Time**: March 9, 2005. 4:10-5 pm, room 1431.

**Speaker**: Fumiko Futamura, Vanderbilt University.

**Title**: On Localized Frames.

**Abstract**: The concept of localization for frames was introduced independently by two groups for two different purposes: one was concerned with constructing Banach frames for particular Banach spaces associated to a particular Riesz basis and the other with understanding the density of frames, and how this relates to their excess. In an effort to unify their conclusions, we introduce a more generalized notion of localization. This notion, in the case of l1-self localization, comes with a natural equivalence class structure.

**Time**: March 2, 2005. 4:10-5 pm, room 1431.

**Speaker**: Tatyana Sorokina, The University of Georgia, Athens.

**Title**: An Octahedral $C^2$ Macro-Element.

**Abstract**: (joint project with Ming-Jun Lai,The University of Georgia, Athens) A macro-element of smoothness $C^2$ is constructed on the split of an octahedron into eight tetrahedra. This new element complements those recently constructed $ Clough-Tocher and Worsey-Farin splits of a tetrahedron by L.L. Schumaker, and P. Alfeld. The new element can be used to construct convenient super-spline spaces with stable local bases and full approximation power that can be used for solving boundary-value problems and $

**Time**: February 15, 2005. 4:10-5 pm, room 1431.

**Speaker**: Akram Aldroubi, Vanderbilt University.

**Title**: Robustness of sampling and reconstruction and Beurling-Landau-type theorems for shift invariant spaces.

**Abstract**: Beurling-Landau-type results are known for a rather small class of functions limited to the Paley-Wiener space and certain spline spaces. Here, we show that the sampling and reconstruction problem in shift invariant spaces is robust with respect to the probing measures as well as to the underlying shift invariant space. As an application we enlarge the class of functions for which a Beurling-Landau-type results hold.

**Time**: February 8, 2005. 4:10-5 pm, room 1431.

**Speaker**: Maxym Yattselev, Vanderbilt University.

**Title**: AAK Theory and its Application to the “Crack” Problem.

**Abstract**:

**Time**: February 1, 2005. 4:10-5 pm, room 1431.

**Speaker**: Andras Kroo, Hungarian Academy of Sciences.

**Title**: On Density of Multivariate Homogeneous Polynomials.

**Abstract**: The classical Weierstrass Theorem states that every function continuous on an interval can be uniformly approximated by algebraic polynomials. This was the first significant density result in Analysis which inspired numerous generalizations applicable to other families of functions. The famous Stone-Weierstrass Theorem gave an extension to subalgebras of C(K), yielding, in particular, the density of multivariate algebraic polynomials. In this talk we shall discuss the density of a special important class of polynomials: the multivariate homogeneous polynomials. Homogeneous polynomials appear in many areas of Analysis. This family is nonlinear, so its density cannot be handled by the Stone-Weierstrass Theorem. In this talk we shall present some recent developments in solving the density problem for homogeneous polynomials.

**Time**: January 25, 2005. 4:10-5 pm, room 1431.

**Speaker**: David Benko, Western Kentucky University.

**Title**: Weighted polynomials on the real line.

**Abstract**: We will consider weighted polynomials of the form $w(x)^n P_n(x)$ where $w(x)$ is a non-negative fixed weight. Professor Saff introduced the problem of finding the uniform closure of these weighted polynomials. In particular the Saff conjecture also arose from him. It was a long standing conjecture for a special class of weights which was finally proved by Professor Totik. In the talk we will give a possible extension of the problem.

**Time**: January 18, 2005. 4:10-5 pm, room 1431.

**Speaker**: Akram Aldroubi, Vanderbilt University.

**Title**: Convolution, average sampling, and Calderon resolution of the identity.

**Abstract**:

## 2004

**Time**: November 17, 2004. 4:10-5 pm, room 1431.

**Speaker**: Paul Leopardi, University of New South Wales, Australia.

**Title**: An equal-measure partition of S^d.

**Abstract**: A construction is given for an equal-measure partition of the unit sphere $S^d \subset R^{d+1}$ called the Recursive-Zhou-Saff-Sloan partition. For $d <= 8$ it can be proven that there is a constant $K_d$ such that, for the RZ partition of $S^d$ into N regions, each region has Euclidean diameter at most $K_d N^{-1/d}$.

**Time**: November 10, 2004. 4:10-5 pm, room 1431.

**Speaker**: Yuliya Babenko, Vanderbilt University.

**Title**: On existence of a function with prescribed norms of its derivatives.

**Abstract**: In this talk we shall discuss the following problem which was posed by Kolmogorov: For given integer $d$, given numbers $M_{\nu_i}$, %$1\leq p_i\leq \infty$ and $1\leq \nu_i \leq r$, $1 \leq i \leq d$ and function space $X$ find necessary and sufficient conditions for existence $x\in X$ such that $$ \left\| x ^ {\left( \nu_i\right) }\right\| _{\infty}= M_{\nu_i}. $$ We shall give a short review of known results and present new ones. In particular, we will give a complete characterization of sets of four numbers such that there exists $l$-monotone function with prescribed smoothness that has these numbers as values of sup-norms of its corresponding derivatives. Along with mentioned classical Kolmogorov problem we shall consider the following related question: if we fix any three out of four given derivatives of order $0<k_1<k_2<r$, what can be said about the remaining one?

**Time**: November 3, 2004. 4:10-5 pm, room 1431.

**Speaker**: Maxim Yattselev, Vanderbilt University.

**Title**: A Remez-Type Theorem for Homogeneous Polynomials. (Joint work with A. Kroo and E.B. Saff).

**Abstract**: In this presentation we are going to consider a problem of estimating of the supremum norm of a polynomials on some set when its norm on a smaller subset is known. This problem was solved by Remez for the interval case. Later A. Kroo and D. Schmidt generalized it for the multivariate polynomials on domains with different smoothness of the boundary. We have considered this problem for class of homogeneous polynomials. In this case a better estimate can be achieved due to their special structure.

**Time**: October 27, 2004. 4:10-5 pm, room 1431.

**Speaker**: Sergiy Borodachov, Vanderbilt University.

**Title**: On minimization of the Riesz s-energy on rectifiable sets.

**Abstract**: In this presentation we are going to consider a problem of estimating of the supremum norm of a polynomials on some set when its norm on a smaller subset is known. This problem was solved by Remez for the interval case. Later A. Kroo and D. Schmidt generalized it for the multivariate polynomials on domains with different smoothness of the boundary. We have considered this problem for class of homogeneous polynomials. In this case a better estimate can be achieved due to their special structure.

**Time**: October 6, 2004. 4:10-5 pm, room 1431.

**Speaker**: Mike Neamtu, Vanderbilt University.

**Title**: Bivariate B-splines Used as Basis Functions for Data Fitting.

**Abstract**: We present results summarizing the utility of bivariate B-splines for solving data fitting problems on bounded domains. These basis functions are defined by certain collections of points in the plane, called knots. The linear span of these functions gives rise to a spline space with good approximation properties. Our numerical results show that the B-splines basis also entertains excellent spectral properties, rendering the B-splines useful for, among other things, iterative solution of data fitting and collocation problems in computational electromagnetics.

**Time**: September 29, 2004. 4:10-5 pm, room 1431.

**Speaker**: G. Lopez Lagomasino, Universidad Carlos III de Madrid, Spain.

**Title**: Ratio asymptotics of Hermite-Pade orthogonal poltnomials for Nikishin systems.

**Abstract**: Multiple orthogonal polynomials share orthogonality relations with a system of measures. They arise naturally when considering simultaneous interpolating rational approximations to a system of analytic functions, and the interpolation conditions are distributed between the different functions. We consider so-called Nikishin systems of functions which are made up of certain types of Cauchy transforms of Borel measures supported on a same finite interval $\Delta$ of the real line, and multiple orthogonal polynomials with respect to the measures generating the Nikishin system with orthogonality “nearly” equally distributed between the different measures. We prove that the ratio of “consecutive” multiorthogonal polynomials converge to an analytic function uniformly on the compact subsets of $C \setminus \Delta$ if the Radon-Nikodym derivative of the measures is $> 0$ a.e. on $\Delta$. This result extends a well known Theorem due to E. A. Rakhmanov.

**Time**: September 22, 2004. 4:10-5 pm, room 1431.

**Speaker**: Larry L. Schumaker, Vanderbilt University.

**Title**: Smooth Macro-Elements on Powell-Sabin-12 Splits.

**Abstract**: For all r >= 0, a family of macro-element spaces of smoothness Cr is constructed based on the Powell-Sabin-12 refinement of a triangulation. These new spaces complement the macro-element spaces based on Powell-Sabin-6 splits which have recently been developed. These new superspline spaces have stable local bases and full approximation power, and can be used to solve boundary-value problems and interpolate Hermite data.

**Time**: September 8, 2004. 4:10-5 pm, room 1431.

**Speaker**: Doug Hardin, Vanderbilt University.

**Title**: Properties of minimum Riesz energy point sets on rectifiable manifolds.

**Abstract**: For a compact set $A\subset {\bf R}^{d’}$, we consider minimal $s$-energy arrangements of $N$ points that interact through a power law (Riesz) potential $V=1/r^{s}$, where $s>0$ and $r$ is Euclidean distance in ${\bf R}^{d’}$. For example, this is the classical Thomson problem of distributing electrons on a sphere in the case $A$ is the unit sphere in ${\bf R}^3$, and $s=1$. In applications one is often interested in determining when such point sets are “uniformly” distributed on $A$ for large $N$. Physicists are also interested in “universal” (i.e. independent of $s$) properties of such configurations. In this talk I will present recent results characterizing asymptotic (as $N\to \infty$) properties of $s$-energy optimal $N$-point configurations for a class of rectifiable $d$-dimensional manifolds and $s\ge d$. This is joint work with E. B. Saff.

**Time**: April 7, 2004. 4:10-5 pm, room 1431.

**Speaker**: Bernd Mulansky, Technical Univ. of Clausthal, Germany.

**Title**: Delaunay configurations.

**Abstract**: Delaunay configurations can be used to select collections of knot-sets in the construction of multivariate spline spaces from simplex spline. We consider geometric and combinatorial properties of Delaunay configurations of a finite point set in the plane, including their efficient computation. Decisive is an interpretation of Delaunay configurations in terms of a convex hull.

**Time**: March 31, 2004. 4:10-5 pm, room 1431.

**Speaker**: Johan de Villiers, University of Stellenbosh, South Africa.

**Title**: On refinable functions and subdivisions with positive masks.

**Abstract**: We present some extensions of the existing theory of refinement equations with positive masks. In particular, attention is given to the geometric converegnce rate of both the cascade algorithm and the subdivision scheme, as well as the sequence space on which the subdivision converges. Finally, we consider the regularity (or degree of smoothness) of the underlying refinable function.

**Time**: March 24, 2004. 4:10-5 pm, room 1431.

**Speaker**: Frank Zeilfelder, University of Mannheim.

**Title**: Approximation and Visualization of Huge Volume Data Sets by Trivariate Splines.

**Abstract**: In recent years, the reconstruction of volume data became a very active area of research since it is important for many general applications such as for instance in scientific visualization and medical imaging. It is known to be a difficult problem to keep all the practical requirements simultaneously into account: high quality visual appearance of the reconstructed objects, quick computation which aims towards the general goal of interactive frame rates, optimal approximation properties of the model and its gradients, insensitiveness for noisy data, efficiency in representation and evaluation of the models. We develop new models for the reconstruction problem of volume data. These models are trivariate splines, i.e. piecewise polynomial functions defined w.r.t. appropriate tetrahedral partitions of the volumetric domain. The talk is subdivided into two parts. In the first part we give some theoretical background on the complex structure of the trivariate splines, while in the second part we show how to turn these results into practical methods for volume data approximation and visualization. Numerical tests show the efficiency of the methods.

**Time**: March 17, 2004. 4:10-5 pm, room 1431.

**Speaker**: Ursula Molter, University of Buenos Aires.

**Title**: Thin and thick Cantor sets.

**Abstract**: In this talk we will discuss the construction of Cantor sets (on the line) associated to summable sequences of positive terms. We will show that to each such Cantor set we can associate an appropriate function h, such that the Hausdorff-h measure of the set is positive.

**Time**: March 3, 2004. 4:10-5 pm, room 1431.

**Speaker**: Doug Hardin, Vanderbilt University.

**Title**:Discrete minimum energy problems on rectifiable manifolds.

**Abstract**:

**Time**: February 5, 2004. 4:10-5 pm, room 1431.

**Speaker**: Andras Kroo, Alfred Renyi Mathematical Institute, Hungarian Academy of Sciences.

**Title**: Uniform norm estimation for factors of multivariate polynomials II.

**Abstract**: We shall consider the following problem of norm estimation of factors of polynomials: given a polynomial p which factors into the product of 2 polynomials p=rq give an upper bound for the norms of factors r and q if the norm of p is known. This problem has been considered in various norms by many authors, it has applications in Banach space theory, number theory, constructive function theory, etc. In this talk we shall discuss this question for spaces of multivariate polynomials endowed with uniform norm on some compact set K, and show how the geometry of K effects the corresponding estimates.

**Time**: January 21, 2004. 4:10-5 pm, room 1431.

**Speaker**: Andras Kroo, Alfred Renyi Mathematical Institute, Hungarian Academy of Sciences.

**Title**:Uniform norm estimation for factors of multivariate polynomials.

**Abstract**: We shall consider the following problem of norm estimation of factors of polynomials: given a polynomial p which factors into the product of 2 polynomials p=rq give an upper bound for the norms of factors r and q if the norm of p is known. This problem has been considered in various norms by many authors, it has applications in Banach space theory, number theory, constructive function theory, etc. In this talk we shall discuss this question for spaces of multivariate polynomials endowed with uniform norm on some compact set K, and show how the geometry of K effects the corresponding estimates.

## 2003

**Time**: December 10, 2003. 4:10-5 pm, room 1431.

**Speaker**: Wolfgang Dahmen, Institut f?r Geometrie und Praktische Mathematik.

**Title**: Adaptive application of operators in wavelet coordinates.

**Abstract**:

**Time**: November 19, 2003. 4:10-5 pm, room 1431.

**Speaker**: Allan Pinkus, Technion.

**Title**: Herman Muntz, 1884-1956.

**Abstract**: The Muntz Theorem is a central theorem in approximation theory. But who was Muntz? How did he come to prove this theorem? In this talk we consider this forgotten mathematician and the odyssey of his life.

**Time**: November 5, 2003. 4:10-5 pm, room 1431.

**Speaker**: Allan Pinkus, Technion.

**Title**: Negative Theorems in Approximation Theory.

**Abstract**: Approximation theory is concerned with the ability to approximate functions and processes by simpler and more easily calculated objects. However there are very definite and intrinsic limitations on approximation processes. In this talk I will survey some of these limitations. Little to no approximation theory background is needed.

**Time**: October 29, 2003. 4:10-5 pm, room 1431.

**Speaker**: Pencho Petrushev, U. South Carolina.

**Title**: Nonlinear n-term approximation from hierarchical spline bases.

**Abstract**: Nonlinear n-term approximation from sequences of hierarchical spline bases generated by multilevel nested triangulations in R2 will be discussed. The emphasis will be placed on the smoothness spaces (B-spaces) governing the rates of nonlinear n-term approximation. The properties of the corresponding Franklin systems will be given as well. It will be explained how the general Jackson-Bernstein machinery can be utilized for characterization of the rates of nonlinear n-term approximation. Also, it will be shown that the B-spaces can be used in the design of algorithms which capture the rate of the best n-term spline approximation. Some related topics and open problems will be discussed as well.

**Time**: October 15, 2003. 4:10-5 pm, room 1431.

**Speaker**: Akram Aldroubi, Vanderbilt University.

**Title**: Wavelet frames on irregular grids, with arbitrary dilation matrices, and in multi-dimension.

**Abstract**: This talk will be introductory and should be understandable by all. We will first introduce the concepts of wavelet bases and wavelet frames. Then, using a one dimensional simple example, we will present the main ideas on how to construct wavelet frames on irregular lattices, and with arbitrary dilation matrices.

**Time**: October 8, 2003. 4:10-5 pm, room 1431.

**Speaker**: Peter Dragnev, Indiana University-Purdue University, Fort Wayne.

**Title**: On a discrete Zolotarev problem with applications to the Alternating Direction Implicit (ADI) method.

**Abstract**: In this talk I will consider a discrete version of the Third Zolotarev Problem. This problem arises in the investigation of optimal parameters of the ADI method for solving partial differential equations. The asymptotics of these parameters are governed by a constrained energy problem for signed measures.

**Time**: September 24, 2003. 4:10-5 pm, room 1431.

**Speaker**: Oleg Davydov, Univ. of Giessen, Germany.

**Title**: Multilevel Bivariate Splines.

**Abstract**: We discuss various possibilities to construct multilevel spline bases in two variables as well as some applications, including recent hierarchical Riesz basis for Sobolev spaces H2(O) on arbitrary polygonal domains.

**Time**: September 18, 2003. 4:10-5 pm, room 1431.

**Speaker**: Peter Alfeld, University of Utah.

**Title**: Trivariate Spline Spaces on Tetrahedral Partitions.

**Abstract**: We consider spaces of smooth piecewise polynomial functions defined on a tetrahedral partition of a three dimensional domain. These spaces can be described in terms of minimal determining sets, i.e., sets of points in the domain that correspond to a set of coefficients which can be chosen arbitrarily and which uniquely determine a spline. The talk will focus on a software package that enables the computation of dimensions and the design of finite elements. The code grew out of a similar package for bivariate splines that has proved instrumental in deriving a number of results in two dimensions.

**Time**: September 10, 2003. 4:10-5 pm, room 1431.

**Speaker**: Andrei Martinez Finkelshtein.

**Title**: Strong asymptotics of Jacobi polynomials with varying nonstandard parameters.

**Abstract**: