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Mathematics & Statistics

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On the convergence of Hermite-Padé approximants

Sergio Medina Peralta

Group of Applied Mathematical Analysis

Carlos III University, Madrid

4:00pm-5:00pm

CMC 130

**Abstract**

Since 1873, when Charles Hermite publishes his proof of the transcendence of e making use of the Hermite-Padé approximants of one system of exponentials the theory of simultaneous rational approximations have been an invaluable tool in many mathematical fields. In addition to number theory, these approximants are connected with the theory of orthogonal polynomials, Riemann-Hilbert problems, models coming from random matrix theory, biorthogonal ensembles, and partial differential equations among others. In this talk we will present some recent results about the uniform convergence of these Hermite-Padé approximants and one application related with the solutions of the Degasperis-Procesi differential equations.

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The Symbols of Unbounded Toeplitz Operators

Joel A. Rosenfeld

University of Florida

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk we will discuss the problem of recovering the symbol of a densely defined Toeplitz operator over the Hardy space. In the bounded setting, there is a clear identification between bounded Toeplitz operators and L-infinity functions on the circle. However, for unbounded Toeplitz operators, there is no straightforward identification. Several different classes of unbounded Toeplitz operators will be considered, and a new method of symbol recovery through the so-called Sarason sub-symbol will be presented. This talk is based on a recently accepted manuscript in the *Journal of Mathematical Analysis and Applications*.

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Subnormality in the Unbounded Case and the Unbounded Bergman Operators

Sherwin Kouchekian

4:00pm-5:00pm

CMC 130

**Abstract**

A short review of the concept of unbounded subnormal operators will be given followed by the obtained known results regarding their self-commutarors. We will then focus on Bergman operators on unbounded regions and give some new results with respect to cyclicity in the unbounded case.

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Rubel's problem on bounded analytic functions

Arthur Danielyan

4:00pm-5:00pm

CMC 130

**Abstract**

The purpose of this talk is to show that for any \(G_\delta\) set \(F\) of Lebesgue measure zero on the unit circle \(T\) there exists a bounded analytic function \(f\) in the unit disc such that the radial limits of \(f\) exist at each point of \(T\) and vanish precisely on \(F\). This solves a problem proposed by L. Rubel in 1973.

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Asymptotic zero distribution of 2D orthogonal polynomial under a small perturbation of the Gaussian case

Seung-Yeop Lee

4:00pm-5:00pm

CMC 130

**Abstract**

We study the asymptotics of 2D orthogonal polynomial when we perturb the Gaussian weight by the small log singularity. In the previous literature, it has been observed that the zeros accumulate on the “mother body”. Also known is that the “mother body” can behave discontinuously under the perturbation of the underlying domain. We study the behavior of zeros when the mother body behaves discontinuously. This is a work in progress with Meng Yang.

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Minimal zeros of optimal approximants in Dirichlet spaces and Jacobi matrices

Catherine Bénéteau

4:00pm-5:00pm

CMC 130

**Abstract**

In this talk, I will shed some light on the structure of the zeros of some polynomials that solve a particular optimization problem in Dirichlet-type spaces indexed by a real parameter \(\alpha\). Given a function \(f\) in a Dirichlet-type space, the optimal approximants in question are polynomials \(p_n\) that minimize the Dirichlet-type norm of \(p(z) f(z)-1\) over all polynomials \(p\) of degree at most \(n\). In previous analysis seminar talks, it has been shown that these optimal approximants are related to certain orthogonal polynomials and reproducing kernels in weighted Dirichlet spaces. In particular, if the index \(\alpha\) of the Dirichlet space is greater than or equal to \(0\) (such as the Hardy space, or the classical Dirichlet space), then all the zeros of the optimal approximants lie outside the closed unit disk, while if the index is negative (such as for the Bergman space), the zeros can creep into the disk. In this talk, I will discuss what is known about how far these zeros can come into the disk and about how they accumulate, and the connection between this extremal problem and the norms of certain Jacobi matrices.

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Universal Taylor series, boundary behaviour and potential theory

Myrto Manolaki

University College, Dublin

4:00pm-5:00pm

CMC 130

**Abstract**

Universality is an abstract notion which relates to various mathematical contexts, ranging from classical to contemporary areas in Mathematics. Generally speaking, an object is called universal if, via a countable process, it can approximate every object in some universe under study. This talk is concerned with one of the most widely-studied instances of universality: the case of universal Taylor series of holomorphic functions, where the approximation is obtained by considering subsequences of partial sums. In recent years, central questions about universal Taylor series have been addressed using tools from Potential Theory. In this talk we will discuss some of these results and we will focus on the boundary behaviour of such functions. In particular, we will discuss the relationship between the boundary behaviour of holomorphic functions on the unit disc and the behaviour of their partial sums on the unit circle. The key-ingredient of our main theorem lies on a new result on harmonic measure, which is of independent interest and can be applied to various classes of universal series.

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On a problem of Pinkus and Wajnryb regarding density of multivariate polynomials

Tom McKinley

4:00pm-5:00pm

CMC 130

**Abstract**

This talk will be based on a recent result answering two questions posed by Allan Pinkus and Bronislaw Wajnryb regarding the density of certain classes of multivariate polynomials.

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Characterization of Stahl compacta and a conjecture on the asymptotics of Hermite-Padé polynomials

E. A. Rakhmanov

4:00pm-5:00pm

CMC 130

**Abstract**

The problem of asymptotics of Hermite-Padé polynomials is one of the hard contemporary open problems of complex analysis. This talk will outline a possible approach towards a resolution of this problem.

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Angle-free methods for boundary element problems on the sphere: The quadrature basis for spherical harmonics

David M. Rogers

USF Chemistry

4:00pm-5:00pm

CMC 130

**Abstract**

The classical moment problem is to classify the set of measures, \(\mu(x)\), that give rise to prescribed moment integrals, \(\int m(x) d \mu(x)\). Potential theory is concerned with calculating solutions to Poisson's equation for a given distribution of source charges (which may be negative). When the sources are confined to a spherical region, it often uses a moment expansion of the charge distribution in terms of monopoles, dipoles, quadrupoles, etc. We prove that all Cartesian moments of such a charge distribution can be reproduced with a point-charge representation on the boundary of a sphere. This boundary representation is equivalent to the usual spherical harmonic expansion without using spherical polar coordinates or complex coefficients. The representation reveals a deep connection between multipolar expansions and boundary element problems in potential theory. [Ref: http://dx.doi.org/10.1063/1.4907404]

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Ludwig Bieberbach and Hilbert’s Eighteenth

Greg McColm

4:00pm-5:00pm

CMC 130

**Abstract**

In 1900, David Hilbert asked the following question: “Is there in n-dimensional Euclidean space also only a finite number of essentially different kinds of groups of motions with a fundamental region?” During the Nineteenth century, much of the effort in what came to be called “mathematical crystallography” was devoted to answering this question affirmatively for \(n=3\) (and, as a sidebar, \(n=2\)). Hilbert’s question was answered by Ludwig Bieberbach a decade later, who also generalized a critical preliminary: what do these groups of motions look like? Since then, these results have been ... upgraded ... using increasingly sophisticated (and inaccessible) mathematical machinery. We ask the question: what do the proofs in Erdös’ book look like?

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A functional equation

Vilmos Totik

4:00pm-5:00pm

CMC 130

**Abstract**

A functional equation containing means is considered. The main problem is to determine if the only solutions are the constant solutions. The answer depends on the continuity and differentiable properties of the components. This is a joint work with Z. Daroczy.

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Cyclic polynomials in anisotropic Dirichlet spaces

Alan Sola

4:00pm-5:00pm

CMC 130

**Abstract**

We characterize polynomials in two complex variables that are cyclic with respect to coordinate shifts acting on weighted Dirichlet spaces in the bidisk.

This reports on joint work with G. Knese (Wash U), L. Kosinski (UJ Krakow), and T. Ransford (Laval).