CCA: Center for Constructive Approximation

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.

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.

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.

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.

Time: December 6, 2005. 4:10-5 pm, room 1431.
Speaker: Casey Leonetti, Vanderbilt University.
Title: Non-Uniform Sampling and Reconstruction From Sampling Sets with Unknown Jitter.
Abstract: This talk will address the problem of non-uniform sampling and reconstruction in the presence of jitter. In sampling applications, the countable set X on which a signal f is sampled is not precisely known. Two main questions are considered. First, if sampling a function f on the countable set X leads to unique and stable reconstruction of f, then when does sampling on the set X', a perturbation of X, also lead to unique and stable reconstruction? Second, if we attempt to recover a sampled function f using the reconstruction operator corresponding to the sampling set X (because the precise sample points are unknown), is the recovered function a good approximation of the original f? Based on work with Akram Aldroubi.

Time: November 29, 2005. 4:10-5 pm, room 1431.
Speaker: Vincent Lunot, INRIA, France.
Title: A Zolotarev Problem with Application to Microwave Filters.
Abstract:

Time: November 15,2005. 4:10-5 pm, room 1431.
Speaker: Dr. Karin Hunter, University of Stellenbosch, South Africa.
Title: A class of symmetric interpolatory subdivision schemes.
Abstract: The well known Dubuc-Deslauriers subdivision masks are symmetric, interpolatory and satisfy a certain polynomial filling property. Here we define a class of symmetric interpolatory masks that include the Dubuc-Deslauriers masks and then give a method to generate masks in this class. We conclude by providing a condition for convergence of a subdivision scheme for a subset of masks in this class.

Time: November 8, 2005. 4:10-5 pm, room 1431.
Speaker: Jorge Stolfi, Institute of Computing, State University of Campinas (Brazil).
Title: Splines on the Sphere (A View from the Other Hemisphere).
Abstract: Polynomial splines on the sphere with triangular topology were defined and thoroughly studied by Alfeld, Neamtu and Schumaker ca. 1996. In this talk we will review the theory of spherical polynomials, their relation to spherical harmonics, and the basics of spherical polynomial spliines. We will then discuss the use of such splines for function approximation and the integration of differential equations on the sphere. (Joint work with Anamaria Gomide)

Time: November 1, 2005. 4:10-5 pm, room 1431.
Speaker: Alex Powell, Vanderbilt University.
Title: Analog to digital conversion for finite frame expansions.
Abstract: We shall dicuss the mathematical aspects of analog-to-digital conversion for redundant signal expansions. We restrict ourselves to the case of finite dimensional data, and consider the naturally associated class of signal expansions given by finite frames. Our focus will be on a special class of algorithms, known as Sigma-Delta quantizers, which are related to error diffusion. We explain the basics of Sigma-Delta schemes and point to ongoing directions of research such as error estimates and stability theorems.

Time: October 18, 2005. 4:10-5 pm, room 1431.
Speaker: Prof. Terry P. Lybrand, Vanderbilt University Center for Structural Biology.
Title: Computer simulation of biomacromolecules and complexes.
Abstract: Computational approaches have become indispensable for study of large biological molecules over the past twenty-plus years. It is also possible, at least in principle, to use simulations and other computational techniques to predict structural and thermodynamic properties. In my group, we are interested primarily in equilibrium thermodynamic properties of biomolecules and complexes, so we use statistical mechanical calculations to estimate these properties. Direct calculation of a partition function for these complex systems is not possible, so we utilize simulation methods like molecular dynamics or (less frequently) Monte Carlo to calculate approximate partition functions via ensemble averaging. I will present some general details of our calculations, discuss common problems and limitations we encounter, and highlight some areas where we hopefully can take advantage of recent mathematical developments to improve our calculations.

Time: September 27, 2005. 4:10-5 pm, room 1431.
Speaker: Yuliya Babenko, Vanderbilt University.
Title: On asymptotically optimal methods of approximation by linear and bilinear splines.
Abstract: In this talk we shall present exact asymptotics of the optimal error in different metrics of linear and bilinear spline interpolation of an arbitrary function $f \in C^2([0,1]^2)$. We shall present review of existing results as well as a series of new ones. Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations (in the case of interpolation by linear splines) and non uniform rectangular partitions (in the case of interpolation by bilinear splines).

Time: September 20, 2005. 4:10-5pm, room 1431.
Speaker: Larry Schumaker, Vanderbilt University.
Title: Trivariate $C^r$ Polynomial Macro-Elements.
Abstract: $C^r$ macro-elements defined in terms of polynomials of degree $8r+1$ on tetrahedra are analyzed. For $r=1,2$, these spaces reduce to well-known macro-element spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.

Time: September 13, 2005. 4:10-5pm, room 1431.
Speaker: Kerstin Hesse, Vanderbilt University.
Title: Optimal Cubature on the Sphere.
Abstract: In this talk I will present results from joint work with Ian H.\,Sloan on cubature (or numerical integration) on the unit sphere $S^2$ in Sobolev spaces. We prove that the worst-case error $e(H^s;Q_m)$ of an $m$-point cubature rule $Q_m$ in the Sobolev space $H^s=H^s(S^2)$, $s>1$, has the optimal order $O(m^{-s/2})$. To achieve this we need two results: On the one hand, we show that for any $m$-point cubature rule $Q_m$ the worst-case cubature error satisfies $e(H^s;Q_m)\geq C\,m^{-s/2}$, with a constant $C$ independent of the rule $Q_m$ (lower bound). On the other hand, we derive an upper bound for the optimal order of the worst-case error by identifying an infinite sequence $(Q_m)$ of $m$-point cubature rules (where $m$ is from an infinite set of natural numbers) for which the worst-case cubature error has an upper bound of the order $O(m^{-s/2})$. The results extend in a natural way to the Sobolev spaces $H^s(S^d)$, where $s>d/2$, on spheres $S^d$ of arbitrary dimension $d>2$ (proof of the lower bound by myself and proof of the upper bound jointly with Johann S.\,Brauchart).

Time: April 19, 2005. 4:10-5 pm, room 1206.
Speaker: Doron Lubinsky, Georgia Tech.
Title: Which weights on R admit Jackson theorems?
Abstract: Let W : R ! (0;1) be continuous. Does W admit a Jackson or Jackson-Favard Inequality? That is, does there exist a sequence f´ng1 n=1 of positive numbers with limit 0 such that for 1 · p · 1; inf deg(P)·n k (f ¡ P)W kLp(R)· ´n k f0W kLp(R) for all absolutely continuous f with k f 0W kLp(R) ¯nite? We show that such a theorem is true i® both lim x!1 W (x) Z x 0 W¡1 = 0 and lim x!1Ãsup [0;x] W¡1!Z 1 x W = 0; with analogous limits as x ! ¡1. In particular W (x) = exp (¡jxj) does not admit a Jackson theorem, although it is well known that W (x) = exp (¡jxj®) ; ® > 1, does. We also construct weights that admit an L1 but not an L1 Jackson theorem (or conversely). The talk will be introductory, and might be accessible to those to whom Jackson and Bernstein sound like the directors of a large corporation.

Time: April 5, 2005. 4:10-5 pm, room 1431.
Speaker: Hong-Tae Shim, Visiting Professor, Sun Moon University, South Korea.
Title: On Gibbs phenomenon in wavelet expansions: its history and development.
Abstract: When a function with jump discontinuity is represented by the trigonometric series, one can observe that its graph exhibits overshoot or downshot near the point of discontinuity. This phenomenon is called the Gibbs' phenomenon, which has been recognized for over a century. However, Gibbs phenomenon is not the special quirk of trigonometric series. It has been shown to exist for many natural approximation, e.g., those involving Fourier series and other classical orthogonal expansions. In this talk, brief history and illustrations are given. We mainly focus on Gibbs phenomenon in wavelet expansions and provide a way to go around it.

Time: March 29, 2005. 4:10-5 pm, room 1431.
Speaker: Gitta Kutyniok, Univ. Giessen, Germany.
Title: Density of irregular wavelet systems.
Abstract: Density conditions have recently turned out to be a useful and elegant tool for studying irregular wavelet systems. In this talk we will discuss necessary and sufficient density conditions on the set of parameters for an irregular wavelet system to constitute a frame. In particular, we will derive a necessary condition on the relationship between the affine density, the frame bounds, and the admissibility condition. Several implications of this relationship will be studied. Moreover, we will prove that density conditions can also be used to characterize existence of wavelet frames, thus serving in particular as sufficient conditions.

Time: March 9, 2005. 4:10-5 pm, room 1431.
Speaker: Fumiko Futamura, Vanderbilt University.
Title: On Localized Frames.
Abstract: The concept of localization for frames was introduced independently by two groups for two different purposes: one was concerned with constructing Banach frames for particular Banach spaces associated to a particular Riesz basis and the other with understanding the density of frames, and how this relates to their excess. In an effort to unify their conclusions, we introduce a more generalized notion of localization. This notion, in the case of l1-self localization, comes with a natural equivalence class structure.

Time: March 2, 2005. 4:10-5 pm, room 1431.
Speaker: Tatyana Sorokina, The University of Georgia, Athens.
Title: An Octahedral $C^2$ Macro-Element.
Abstract: (joint project with Ming-Jun Lai,The University of Georgia, Athens) A macro-element of smoothness $C^2$ is constructed on the split of an octahedron into eight tetrahedra. This new element complements those recently constructed $ Clough-Tocher and Worsey-Farin splits of a tetrahedron by L.L. Schumaker, and P. Alfeld. The new element can be used to construct convenient super-spline spaces with stable local bases and full approximation power that can be used for solving boundary-value problems and $

Time: February 15, 2005. 4:10-5 pm, room 1431.
Speaker: Akram Aldroubi, Vanderbilt University.
Title: Robustness of sampling and reconstruction and Beurling-Landau-type theorems for shift invariant spaces.
Abstract: Beurling-Landau-type results are known for a rather small class of functions limited to the Paley-Wiener space and certain spline spaces. Here, we show that the sampling and reconstruction problem in shift invariant spaces is robust with respect to the probing measures as well as to the underlying shift invariant space. As an application we enlarge the class of functions for which a Beurling-Landau-type results hold.

Time: February 8, 2005. 4:10-5 pm, room 1431.
Speaker: Maxym Yattselev, Vanderbilt University.
Title: AAK Theory and its Application to the "Crack" Problem.
Abstract:

Time: February 1, 2005. 4:10-5 pm, room 1431.
Speaker: Andras Kroo, Hungarian Academy of Sciences.
Title: On Density of Multivariate Homogeneous Polynomials.
Abstract: The classical Weierstrass Theorem states that every function continuous on an interval can be uniformly approximated by algebraic polynomials. This was the first significant density result in Analysis which inspired numerous generalizations applicable to other families of functions. The famous Stone-Weierstrass Theorem gave an extension to subalgebras of C(K), yielding, in particular, the density of multivariate algebraic polynomials. In this talk we shall discuss the density of a special important class of polynomials: the multivariate homogeneous polynomials. Homogeneous polynomials appear in many areas of Analysis. This family is nonlinear, so its density cannot be handled by the Stone-Weierstrass Theorem. In this talk we shall present some recent developments in solving the density problem for homogeneous polynomials.

Time: January 25, 2005. 4:10-5 pm, room 1431.
Speaker: David Benko, Western Kentucky University.
Title: Weighted polynomials on the real line.
Abstract: We will consider weighted polynomials of the form $w(x)^n P_n(x)$ where $w(x)$ is a non-negative fixed weight. Professor Saff introduced the problem of finding the uniform closure of these weighted polynomials. In particular the Saff conjecture also arose from him. It was a long standing conjecture for a special class of weights which was finally proved by Professor Totik. In the talk we will give a possible extension of the problem.

Time: January 18, 2005. 4:10-5 pm, room 1431.
Speaker: Akram Aldroubi, Vanderbilt University.
Title: Convolution, average sampling, and Calderon resolution of the identity.
Abstract:

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

Time: November 3, 2004. 4:10-5 pm, room 1431.
Speaker: Maxim Yattselev, Vanderbilt University.
Title: A Remez-Type Theorem for Homogeneous Polynomials. (Joint work with A. Kroo and E.B. Saff).
Abstract: In this presentation we are going to consider a problem of estimating of the supremum norm of a polynomials on some set when its norm on a smaller subset is known. This problem was solved by Remez for the interval case. Later A. Kroo and D. Schmidt generalized it for the multivariate polynomials on domains with different smoothness of the boundary. We have considered this problem for class of homogeneous polynomials. In this case a better estimate can be achieved due to their special structure.

Time: October 27, 2004. 4:10-5 pm, room 1431.
Speaker: Sergiy Borodachov, Vanderbilt University.
Title: On minimization of the Riesz s-energy on rectifiable sets.
Abstract: In this presentation we are going to consider a problem of estimating of the supremum norm of a polynomials on some set when its norm on a smaller subset is known. This problem was solved by Remez for the interval case. Later A. Kroo and D. Schmidt generalized it for the multivariate polynomials on domains with different smoothness of the boundary. We have considered this problem for class of homogeneous polynomials. In this case a better estimate can be achieved due to their special structure.

Time: October 6, 2004. 4:10-5 pm, room 1431.
Speaker: Mike Neamtu, Vanderbilt University.
Title: Bivariate B-splines Used as Basis Functions for Data Fitting.
Abstract: We present results summarizing the utility of bivariate B-splines for solving data fitting problems on bounded domains. These basis functions are defined by certain collections of points in the plane, called knots. The linear span of these functions gives rise to a spline space with good approximation properties. Our numerical results show that the B-splines basis also entertains excellent spectral properties, rendering the B-splines useful for, among other things, iterative solution of data fitting and collocation problems in computational electromagnetics.

Time: September 29, 2004. 4:10-5 pm, room 1431.
Speaker: G. Lopez Lagomasino, Universidad Carlos III de Madrid, Spain.
Title: Ratio asymptotics of Hermite-Pade orthogonal poltnomials for Nikishin systems.
Abstract: Multiple orthogonal polynomials share orthogonality relations with a system of measures. They arise naturally when considering simultaneous interpolating rational approximations to a system of analytic functions, and the interpolation conditions are distributed between the different functions. We consider so-called Nikishin systems of functions which are made up of certain types of Cauchy transforms of Borel measures supported on a same finite interval $\Delta$ of the real line, and multiple orthogonal polynomials with respect to the measures generating the Nikishin system with orthogonality "nearly" equally distributed between the different measures. We prove that the ratio of "consecutive" multiorthogonal polynomials converge to an analytic function uniformly on the compact subsets of $C \setminus \Delta$ if the Radon-Nikodym derivative of the measures is $> 0$ a.e. on $\Delta$. This result extends a well known Theorem due to E. A. Rakhmanov.

Time: September 22, 2004. 4:10-5 pm, room 1431.
Speaker: Larry L. Schumaker, Vanderbilt University.
Title: Smooth Macro-Elements on Powell-Sabin-12 Splits.
Abstract: For all r >= 0, a family of macro-element spaces of smoothness Cr is constructed based on the Powell-Sabin-12 refinement of a triangulation. These new spaces complement the macro-element spaces based on Powell-Sabin-6 splits which have recently been developed. These new superspline spaces have stable local bases and full approximation power, and can be used to solve boundary-value problems and interpolate Hermite data.

Time: September 8, 2004. 4:10-5 pm, room 1431.
Speaker: Doug Hardin, Vanderbilt University.
Title: Properties of minimum Riesz energy point sets on rectifiable manifolds.
Abstract: For a compact set $A\subset {\bf R}^{d'}$, we consider minimal $s$-energy arrangements of $N$ points that interact through a power law (Riesz) potential $V=1/r^{s}$, where $s>0$ and $r$ is Euclidean distance in ${\bf R}^{d'}$. For example, this is the classical Thomson problem of distributing electrons on a sphere in the case $A$ is the unit sphere in ${\bf R}^3$, and $s=1$. In applications one is often interested in determining when such point sets are ``uniformly'' distributed on $A$ for large $N$. Physicists are also interested in ``universal'' (i.e. independent of $s$) properties of such configurations. In this talk I will present recent results characterizing asymptotic (as $N\to \infty$) properties of $s$-energy optimal $N$-point configurations for a class of rectifiable $d$-dimensional manifolds and $s\ge d$. This is joint work with E. B. Saff.

Time: April 7, 2004. 4:10-5 pm, room 1431.
Speaker: Bernd Mulansky, Technical Univ. of Clausthal, Germany.
Title: Delaunay configurations.
Abstract: Delaunay configurations can be used to select collections of knot-sets in the construction of multivariate spline spaces from simplex spline. We consider geometric and combinatorial properties of Delaunay configurations of a finite point set in the plane, including their efficient computation. Decisive is an interpretation of Delaunay configurations in terms of a convex hull.

Time: March 31, 2004. 4:10-5 pm, room 1431.
Speaker: Johan de Villiers, University of Stellenbosh, South Africa.
Title: On refinable functions and subdivisions with positive masks.
Abstract: We present some extensions of the existing theory of refinement equations with positive masks. In particular, attention is given to the geometric converegnce rate of both the cascade algorithm and the subdivision scheme, as well as the sequence space on which the subdivision converges. Finally, we consider the regularity (or degree of smoothness) of the underlying refinable function.

Time: March 24, 2004. 4:10-5 pm, room 1431.
Speaker: Frank Zeilfelder, University of Mannheim.
Title: Approximation and Visualization of Huge Volume Data Sets by Trivariate Splines.
Abstract: In recent years, the reconstruction of volume data became a very active area of research since it is important for many general applications such as for instance in scientific visualization and medical imaging. It is known to be a difficult problem to keep all the practical requirements simultaneously into account: high quality visual appearance of the reconstructed objects, quick computation which aims towards the general goal of interactive frame rates, optimal approximation properties of the model and its gradients, insensitiveness for noisy data, efficiency in representation and evaluation of the models. We develop new models for the reconstruction problem of volume data. These models are trivariate splines, i.e. piecewise polynomial functions defined w.r.t. appropriate tetrahedral partitions of the volumetric domain. The talk is subdivided into two parts. In the first part we give some theoretical background on the complex structure of the trivariate splines, while in the second part we show how to turn these results into practical methods for volume data approximation and visualization. Numerical tests show the efficiency of the methods.

Time: March 17, 2004. 4:10-5 pm, room 1431.
Speaker: Ursula Molter, University of Buenos Aires.
Title: Thin and thick Cantor sets.
Abstract: In this talk we will discuss the construction of Cantor sets (on the line) associated to summable sequences of positive terms. We will show that to each such Cantor set we can associate an appropriate function h, such that the Hausdorff-h measure of the set is positive.

Time: March 3, 2004. 4:10-5 pm, room 1431.
Speaker: Doug Hardin, Vanderbilt University.
Title:Discrete minimum energy problems on rectifiable manifolds.
Abstract:

Time: February 5, 2004. 4:10-5 pm, room 1431.
Speaker: Andras Kroo, Alfred Renyi Mathematical Institute, Hungarian Academy of Sciences.
Title: Uniform norm estimation for factors of multivariate polynomials II.
Abstract: We shall consider the following problem of norm estimation of factors of polynomials: given a polynomial p which factors into the product of 2 polynomials p=rq give an upper bound for the norms of factors r and q if the norm of p is known. This problem has been considered in various norms by many authors, it has applications in Banach space theory, number theory, constructive function theory, etc. In this talk we shall discuss this question for spaces of multivariate polynomials endowed with uniform norm on some compact set K, and show how the geometry of K effects the corresponding estimates.

Time: January 21, 2004. 4:10-5 pm, room 1431.
Speaker: Andras Kroo, Alfred Renyi Mathematical Institute, Hungarian Academy of Sciences.
Title:Uniform norm estimation for factors of multivariate polynomials.
Abstract: We shall consider the following problem of norm estimation of factors of polynomials: given a polynomial p which factors into the product of 2 polynomials p=rq give an upper bound for the norms of factors r and q if the norm of p is known. This problem has been considered in various norms by many authors, it has applications in Banach space theory, number theory, constructive function theory, etc. In this talk we shall discuss this question for spaces of multivariate polynomials endowed with uniform norm on some compact set K, and show how the geometry of K effects the corresponding estimates.

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: IIn 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: