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 cca@vanderbilt.edu or aleksandr.b.reznikov@vanderbilt.edu.

2017

Time:

Speaker:

Title:

Abstract:

2016

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: 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: 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)|
^pdω(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_pω^‘(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^iuπf_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)

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-Szeg� 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 Pad� 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 C¹-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!1�sup [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.

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:

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