Local Informations

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Written by Seminario
Monday, 01 October 2012 00:10

Seminar on Orthogonality, Approximation Theory and Applications

Contact:

Past seminars

Thursday, July 4 2013, 16:00. Room 2.2.D08

Speaker: Alexander G. Ramm, Kansas State University.

Title: Wave scattering by many small particles and creating materials with desired refraction coeficients.

Abstract: Many-body wave scattering problems are solved asymptotically as the size a of the particles tends to zero and the number of the particles tends to infinity. Acoustic, quantum-mechanical, and electromagnetic wave scattering by many small particles are studied. Computational methods for solving many-body wave scattering problems in the case of small scatterers are developed. They allow one to treat wave scattering by as many as 10^6 small particles. This theory allows one to give a recipe for creating materials with a desired refraction coeficient. One can create material with negative refraction, that is, the material in which the group velocity is directed opposite to the phase velocity. One can create material with desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle.

Friday, July 5 2013, 16:00. Room 2.2.D08

Speaker: Maxim S. Derevyagin, Institut fur Mathematik, TU Berlin.

Title: On Darboux transformations and polynomial mappings.

Abstract: Let d\mu(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of d\mu(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. At first, I am going to present a matrix interpretation of the unwrapping transformation from S(\lambda) to \lambda S(\lambda^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it will be shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to the Carlitz problem appears as a shifted Darboux transformation. At second, we consider applications of Darboux transformations to Pade approximation.

Thursday, June 27 2013, 16:00. Room 2.2.D08

Title: CMV matrices and spectral transformations of measures.

Abstract: In this talk we deal with a matrix approach to spectral transformations of nontrivial probability measures supported on the unit circle. In some previous work with L. Garza, when linear spectral transformations are considered, we have analyzed the connection between the Hessenberg matrices  (GGT matrices) associated with the multiplication operator in terms of  bases of orthonormal polynomials corresponding to the initial and the transformed measures, respectively. Here, we deal with the connection between  the CMV matrices  associated with the multiplication operator with respect to bases of Laurent orthonormal polynomials corresponding to a non trivial probability measure supported on the unit circle and a Hermitian linear transformation of it. We focus our attention on the Christoffel and Geronimus transformations, where the QR and Cholesky factorizations play a central role. A comparison with the results about the LU factorization of monic Jacobi matrices is presented as well as some recent results by M. Derevyagin, L. Vinet, and A. Zhedanov based on the connection between CMV matrices and Jacobi matrices based on the Szegö transformation. This is  a joint work with M. J. Cantero, L. Moral and L. Velázquez (Universidad de Zaragoza).

Thursday, June 13 2013, 16:00. Room 2.2.D08

Title: Intricate dynamics in superlinear indefinite problems.

Abstract: In this talk I will present a multiplicity result for positive solutions of a superlinear indefinite differential equation which arises in population dynamics to model the behavior of a species in a polluted habitat where competitive and cooperative effects among the individuals are combined. Using the amplitude of the cooperative effect as a parameter, we have been able to describe the bifurcation diagrams of the solutions in a one dimensional case, which are rather complex due to the high number of the solutions that the problem can admit. The multiplicity in higher dimensions, as well as the linear stability of the solutions will be also discussed in full generality. Finally, I will present some numerical experiments, both for cases already treated analytically and new ones. The content of the talk is based on joint works with J. Lopez-Gomez (UCM), M. Molina-Meyer (UC3M) and F. Zanolin (Università degli Studi di Udine).

Thursday, May 30 2013, 16:00. Room 2.2.D08

Speaker: Ana Isabel Mendes, Instituto Politécnico de Leiria (Portugal).

Title: On the relation between the full Kostant-Toda lattice and matrix orthogonal polynomials.

Abstract: In this work we characterize a full Kostant-Toda system in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to studied the solution of this dynamical system we give explicit expressions for the resolvent functional and we also obtain, under some conditions, a representation of the vector functionals associated with this system.

Speaker: András Kroó, Hungarian Academy of Sciences - Cátedra de Excelencia.

Title: Bernstein-Markov type inequalities on star-like domains in R^d with applications to norming sets.

Abstract: Bernstein-Markov type inequalities widely used in various areas of analysis are related to estimating derivatives of polynomials on various domains. In this talk we shall give a survey of the history of this problem and present some new Bernstein-Markov type inequalities for multivariate polynomials on star like domains. These new inequalities will be shown to be useful in the study of cardinality of norming sets, or admissible meshes. Admissible meshes are applied in various areas, for instance they are used for discrete least squares approximation, extracting discrete extremal sets of Fekete and Leja type, scattered data interpolation, cubature formulas, etc.

Thursday, April 25 2013, 16:00. Room 2.2.D08

Speaker: María del Carmen Reguera, Universidad Autónoma de Barcelona.

Title: Weighted estimates for the Bergman projection and Sarason Conjecture.

Abstract: We will present some partial results on the two weight problem for the Bergman projection on the disc. This problem is connected to a conjecture of Sarason on the composition of Toeplitz operators in the Bergman space, where we will provide a complete characterization. We will also present some counterexamples to the extended Sarason conjecture for the Bergman space. This is joint work with A. Aleman and S. Pott.

Thursday, April 18 2013, 16:00. Room 2.2.D08

Title: Orthogonal polynomials and edge detection.

Abstract: In this talk we propose a novel algorithm that make use of two dimensional discrete orthogonal polynomials for detection of edges in 2D and depth maps images. Also we present some numerical results and compare them with the well known algorithms.

Thursday, April 11 2013, 16:00. Room 2.2.D08

Speaker: Jesús Guillera.

Title: Ramanujan series "upside-down".

Abstract: Series for $1/\pi$ of the following form

$\sum_{n=0}^{\infty}\frac{(s)_n \left(\frac{1}{2}\right)_n (1-s)_n}{(1)_n^3}(a+b n) z^n = \frac{1}{\pi},$

are known as Ramanujan-type formulas.  As usual, $(s)_n=s(s+1)\cdots(s+n-1)$ is Pochhammer's symbol, $s \in \{ 1/2, 1/3, 1/4, 1/6 \}$, and $z$, $a$, and $b$ are algebraic numbers. In $1914$ Ramanujan identified $17$ series of this kind. In this talk we show how to evaluate the related upside-down" series (when $s\ne 1/6$)

$\sum_{n=1}^{\infty}\frac{(1)_n^3}{(s)_n \left(\frac{1}{2}\right)_n (1-s)_n} \frac{a-b n}{n^3} z^{-n},$

in terms of values of the Epstein zeta function

$\sum_{(m,n) \in \mathbb{Z}^2}^{\prime} \frac{1}{(An^2+Bmn+Cm^2)^t},$

at $t=2$. Since Epstein zeta functions often reduce to Dirichlet $L$-values, this recipe leads to examples such as

$\sum_{n=1}^{\infty} \frac{(1)_n^3}{\left( \frac{1}{3} \right)_n \left( \frac{1}{2} \right)_n \left(\frac{2}{3} \right)_n} \frac{4-15n}{n^3} \, (-4)^{-n} = 27 L_{-3}(2).$

Observe that convergent upside-down series are associated to divergent Ramanujan-type series. (Joint work with Mathew Rogers)

Thursday, April 4 2013, 16:00. Room 2.2.D08

Speaker: Alfredo Deaño, KU Leuven.

Title: Complex orthogonal polynomials and Gaussian quadrature.

Abstract: We present an extension of the classical ideas of Gaussian quadrature to integrals in the complex plane. This was originally motivated by the problem of computing oscillatory integrals on the real axis. Because the weight function is generally not real or positive, the existence of the associated family of orthogonal polynomials is not clear a priori, and their zeros are typically distributed along curves in the complex plane. This zero distribution of the roots of these OPs can be analyzed using logarithmic potential theory, and asymptotic information can usually be obtained via Riemann--Hilbert techniques. We present several cases that have been considered recently or that are under study, and (time permitting) we point out possible open problems in this area as well as connections with random matrix theory. This is joint and ongoing work with A. Asheim, D. Huybrechs, A. B. J. Kuijlaars (KU Leuven, Belgium) and P. Román (Universidad Nacional de Córdoba, Argentina).

Thursday, March 21 2013, 16:00. Room 2.2.D08

Speaker: Serhan Varma, Ankara University.

Title: d-Orthogonal polynomials by generating functions and an application to the linear positive operators.

Abstract: We shall introduce the concept of d-orthogonality of a given polynomial sequence. First, we will describe a method for checking d-orthogonality of polynomial sequences defined by generating functions. Then, we will give a generalization of the Laguerre polynomials in the context of d-orthogonality by a generating function of a certain form. In the last section, we will show how certain d-orthogonal polynomials (the Gould-Hopper polynomials) can be used to extend the Szasz operators.

Thursday, March 14 2013, 16:00. Room 2.2.D08

Title: An approach to the interpolatory quadrature formulae.

Abstract: Given a measure, we consider an arbitrary schemes of nodes in the interior of its support. For the corresponding interpolatory quadrature rule we study  convergent conditions in terms of the node's distribution. In this analysis, a connection between Fourier series of orthogonal polynomials and interpolatory quadrature formulae plays a fundamental role.

Thursday, March 7 2013, 16:00. Room 2.2.D08

Title: Convergence of tipe II Hermite-Padé approximants.

Abstract: Let $(s_{1},\ldots, s_{m})= \mathcal{N}(\sigma_{1},\dots, \sigma_{m})$ be a Nikishin system and $\Delta_{1}$ be the convex hull of $\mathit{supp}(\sigma_{1})$. Let $(r_{1},\ldots, r_{m})$ be rational functions such that $r_{k}(\infty)=0$ and the poles of $r_{k}$ lie in $\mathbb{C}\setminus \Delta_{1}$, for all $k=1,\ldots,m$. We study the convergence of the diagonal sequence of type II Hermite-Padé approximants associated to the system of functions $(f_{1},\ldots, f_{m})$ where $f_{k}(z)=\int \frac{ds_{k}(x)}{z-x}+ r_{k}$, $k=1,\ldots,m$.

Thursday, February 28 2013, 16:00. Room 2.2.D08

Speaker: András Kroó, Hungarian Academy of Sciences - Cátedra de Excelencia.

Title: Density of Multivariate Polynomials on Convex and Star like domains.

Abstract: A central question in Approximation Theory concerns the possibility of approximation of continuous functions by various families of polynomials, that is density of classes of polynomials. On one hand the density of a given polynomial family depends on the algebraic structure of this set. In addition, in the multivariate case the question of density is also intricately related to the geometric properties of the underlying domain on which the approximation is studied.
In the present talk we shall explore this interplay between algebraic and geometric properties in the study of density of various families of multivariate polynomials on compact subsets of R^d, in particular convex bodies or star like domains. The families of polynomials will include multivariate homogeneous polynomials, convex polynomials and incomplete polynomials.

Thursday, February 14 2013, 16:00. Room 2.2.D08

Title: Entanglement theory and multipartite maximally entangled states.

Abstract: Quantum information thaory offers revolutionary ways to transmit and process information such as quantum cryptography and quantum computation. Besides its interest in the foundations of quantum theory, entanglement is believed to play a fundamental role in these tasks and it is nowadays regarded as a resource. This has given rise to entanglement theory, which aims at characterizing entangled states, identifying which transformations among them are possible and quantifying how useful they can be. Since I expect most members of the audience not to be familiar with quantum theory, I will devote the first half of my talk to present the rules of the quantum game and to introduce some basic concepts of entanglement theory. In the second half I will explain some recent results of my own characterizing which deterministic conversions among multipartite entangled states are possible. In particular, this allows to identify which states are most useful as a resource: multipartite maximally entangled states. I will try to give a flavour of the mathematical tools that are used, which mainly rely on Matrix Analysis.

Thursday, February 7 2013, 16:00. Room 2.2.D08

Title: Quasi-isometries and isoperimetric inequalities in planar domains.

Abstract: In this work we study the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces endowed with their Poincarée metrics. This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of any non-linear isoperimetric inequality.

Thursday, January 31 2013, 16:00. Room 2.2.D08

Title: On the Banach-Saks property.

Abstract: We will present a soft introduction to the Banach-Saks property in Banach spaces. Recall that a subset A of a Banach space has the Banach-Saks property if every sequence in A has a subsequence whose arithmetic means are convergent. We will survey classical results and provide examples of spaces with and without this property. Finally, we will present recent results on the convex hull of Banach-Saks sets obtained in collaboration with J. López-Abad (CSIC) and C. Ruiz (UCM).

Thursday, January 24 2013, 16:00. Room 2.2.D08

Speaker: Bernhard Beckermann, Université de Lille.

Title: An error analysis for rational Galerkin projection applied to the Sylvester equation.

Abstract: The Sylvester equation AX - XB = C in the unknown matrix X for known matrices A,B,C occur naturally in matrix eigendecompositions, control theory, model reduction, but also numerical solutions of Riccati equations, or image processing. In many of these applications, A,B are fairly large and sparse, and C is of low rank. In this talk we suggest a new error estimate for Galerkin projection on rational Krylov spaces in the case where the numerical ranges of A and B have an empty intersection [1]. As an important ingredient we present an orthogonal decomposition of the residual, where each term can be bounded by considering a matrix-valued extremal problem for rational unctions. For the latter, an approximate solution can be given via Gonchar's rational version of the Bernstein-Walsh inequality [3], and via recent techniques for bounding matrix functions [2].
References:
[1] B. Beckermann: An error analysis for rational Galerkin projection applied to the Sylvester equation, SIAM J. Num. Anal. 49 (2012), 2430-2450.
[2] B. Beckermann, Image numérique, GMRES et polynômes de Faber, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 855-860.
[3] A.A. Gonchar, Zolotarev problems connected with rational functions. Mathem. Digest (Matem. Sbornik), 7(4) (1969) 623-635.

Thursday, January 17 2013, 16:00. Room 2.2.D08

Title: Analysis on the behaviour of the solutions of several elliptic equations in domains with a multiple crack section.

Abstract: As the main problem, the bi-Laplace equation is considered (being a model from fracture mechanics/elasticity) in a bounded domain in R^2 (basically a ball around the origin) and inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary of the domain. In addition, there is a finite collection of curves modeling a multiple crack formation, focusing at the origin. Possible types of the behaviour of solution at the tip of such admissible multiple cracks, being a singularity boundary point, are described, on the basis of blow-up scaling techniques and spectral theory of pencils of non-self-adjoint operators. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of harmonic polynomials, which are now represented by pencil eigenfunctions, instead of their classical representation via a standard Sturm--Liouville problem. Eventually, for a fixed admissible crack formation at the origin, this allows us to describe all boundary data, which can generate such a blow-up crack structure. In particular, we will discuss how the co-dimension of this data set increases with the number of asymptotically straight-line cracks focusing at 0. As a by-product, similar problems of multiple cracks focusing at zero are studied for the Laplace and p-Laplace equations. In the latter case, nonlinear eigenfunctions of the rescaled equation in blow-up" variables are obtained via branching from harmonic polynomials.

Thursday, January 10 2013, 16:00. Room 2.2.D08

Title: Finite operators and Foelner sequences.

Abstract: In this talk I will analyze Foelner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. I will show that several classes of operators have Foelner sequences and also study the structure of operators with that have no Foelner sequence. Time permitting I will address some applications to spectral approximation problems and give an intrinsic characterization of operator algebras having a Foelner sequence in terms of completely positive maps.

Thursday, December 20 2012, 16:00. Room 2.2.D08

Title: Orthogonal polynomials with respect to differential operators and matrix orthogonal polynomials.

Abstract: This  work  deals with the concept of orthogonal polynomials with respect to a differential operator, the study of the strong asymptotic behavior of eigenpolynomials of exactly solvable operators, and matrix orthogonal polynomials. We consider  orthogonal polynomials with respect to either a Jacobi, Laguerre or Hermite operator and a finite positive Borel measure  $\mu$  satisfying certain conditions. For a positive integer $m$, we analyze the  conditions over the measure $\mu$ in order to guarantee the existence of an infinite  sequence of monic polynomials $\{Q_n\}_{n=m+1}^{\infty}$, where each $Q_n$ has degree $n$ and  orthogonal with respect to the operator. We   consider algebraic and analytic  properties of this sequence.  A fluid dynamics  model  for the interpretation of the zeros of these polynomials is also considered.

Some of the results obtained for a classical operator are generalized  by  considering orthogonal polynomials with respect to a wider class of linear differential operators. We  analyze the uniqueness and zero location of these polynomials. An interesting phenomena  occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator we also find a classification, in terms of a  system of difference equations with varying coefficients,  of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials.  We also obtain a curve which contains the set of accumulation points of the zeros of these polynomials for the case of a first order differential operator giving also the strong asymptotic behavior.

We consider as well the study of the strong asymptotic behavior the eigenpolynomials of exactly solvable operators.  Under the assumption that the leading coefficient of the operator is a real polynomial, we obtain a formula for the strong asymptotic behavior of the eigenpolynomials   on certain compact subsets of the complex plane. As an application, we study the strong asymptotic behavior of a sequence of monic orthogonal polynomials with respect to a Sobolev inner product  which are eigenfunctions of a fourth order differential operator.

It is also object of study  a new class of matrix orthogonal polynomials of arbitrary size satisfying a second order matrix differential equation. For matrix polynomials of size 2, we find an explicit expression of the sequence of orthonormal polynomials with respect to a weight by using a Rodrigues' formula for these polynomials. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.

Thursday, December 13 2012, 16:00. Room 2.2.D08

Speaker: Andrei Martínez-Finkelshtein, Universidad de Almería.

Title: Phase transitions and equilibrium measures.

Abstract: Many asymptotic results in random matrix theory are best formulated in terms of the equilibrium measures in an external field; the support of such a measure is one of the main parameters of the problem, since it is the accumulation set of the eigenvalues of these matrices (or equivalently, of the zeros of the associated orthogonal polynomials). In a rather broad class of problems the potential (external field) is a polynomial. Its coefficients correspond to some physical aspects of he underlying model, and their variation affect naturally the structure of the support of the equilibrium measure. The transitions from one configuration of the support to another reflect the phase transitions in the physical model, and are object of the main study by the physicists. In this talk we discuss how the changes in the coefficients of the external field are associated to the evolution of the support of the equilibrium measure, and what are the mechanisms of the phase transitions. These results are illustrated in the case of a quartic potential, which is the first non-trivial and sufficiently representative case. This is a joint work with E.A. Rakhmanov and R. Orive.

Thursday, November 29 2012, 16:00. Room 2.2.D08

Speaker: Dixan Peña, Ghent University.

Title: On the generation of special monogenic functions using Fueter's theorem.

Abstract: Clifford analysis deals with partial differential operators that arise naturally within the context of a Clifford algebra. The most fundamental one is the generalized Cauchy-Riemann operator and its null solutions are called monogenic functions. Fueter's theorem discloses a remarkable connection existing between holomorphic functions and monogenic functions. The aim of our talk is to provide a short overview of this subject and to present some examples of special monogenic functions generated by this technique.

Thursday, November 22 2012, 16:00. Room 2.2.D08

Title: Approximation properties of shift-invariant spaces and the spectral function.

Abstract: Consider the operator $\mathcal{D}_Af(\cdot):=|\det(A)|^{\frac{1}{2}}\,f(A\,\cdot)$ ($f\in L^2(\mathbb{R}^d)$) associated to a given dilation A on $\mathbb{R}^d$. Firstly we will review the role played by A-sets (measurable sets such that A(G)=G) in the characterization of A-reducing spaces, which are shift-invariant spaces satisfying $\mathcal{D}_AV=V$. We will also introduce the notions of (G,A)-density point of a set, (G,A)-approximate continuity point of a measurable function and function (G,A)-locally non zero at a point. We will also study the A-approximation and A-density orders of shift-invariant spaces, obtaining some results which generalize some theorems due to De Boor, DeVore and Ron, Bownik and Rzeszotnik, and San Antolín.
We will also introduce and study the wider notions of (G,A)-approximation and (G,A)-density orders, which allow us to develope an approximation theory in A-reducing spaces. All the provided conditions focus on the local behaviour at the origin of the spectral function $\sigma_V$ of V.

Thursday, November 15 2012, 16:00. Room 2.2.D08

Title: Muckenhoupt inequality with three measures and Sobolev orthogonal polynomials.

Abstract: We generalize the Muckenhoupt inequality from two measures to three under certain conditions. This implies a very simple characterization of the boundedness of the multiplication operator and thus the boundedness of the zeros and the asymptotic behavior of the Sobolev orthogonal polynomials, for a large class of measures which includes almost every example in the literature.

Thursday, November 8 2012, 16:00. Room 2.2.D08

Title: Aproximación simultánea vinculada al estudio de la irracionalidad de los valores de la función zeta de Riemann.

Abstract: Uno de los problemas abiertos más interesantes de la teoría de números versa sobre la naturaleza aritmética de los valores de la función zeta de Riemann (z(s), Re s>1) en los enteros positivos 2,3,... Roger Apéry, en el año 1979, dio una demostración de la irracionalidad del número z(3) (en la actualidad denominada constante de Apéry). Dicho resultado, en un principio algo polémico, inspiró a  varios matemáticos que construyeron diferentes métodos para explicar la irracionalidad de dicha constante. Sorprendentemente, estos métodos conducen a la misma sucesión de aproximantes racionales (denominados aproximantes de Apéry). En la primera parte de esta charla presentaremos una nueva sucesión de aproximantes Diofánticos de la constante z(3). Esta sucesión se obtiene a partir de un problema de aproximación racional simultánea asociado a un problema de Riemann-Hilbert indeterminado. A continuación mostraremos que la nueva sucesión de aproximantes racionales a z(3), que además prueba su irracionalidad, dará lugar a una relación de recurrencia, así como a un nuevo desarrollo en fracciones continuas para z(3). Posteriormente, se expondrán infinitos aproximantes Diofánticos que dependen de ciertos parámetros enteros. Luego se vinculará cada uno de estos aproximantes con un problema de aproximación racional simultánea en el infinito. De esta manera, siguiendo la misma estrategia, se deducirán relaciones de recurrencia, así como nuevos desarrollos en fracciones continuas para z(3).
Las técnicas empleadas en el estudio de la constante de Apéry nos han permitido introducir ciertas funciones racionales (deducidas a partir de un problema de aproximación racional simultánea) a partir de las cuales se infieren nuevos aproximantes racionales a z(4). También deducimos relaciones de recurrencia que permiten obtener de forma directa nuevos desarrollos en fracciones continuas para z(4). Estos nuevos aproximantes mejoran la convergencia respecto de algunos resultados previos obtenidos por otros autores.
Por último, se plantea un problema general de aproximación racional simultánea en el infinito, a partir del cual, se consiguen buenos aproximantes racionales a los valores de la función zeta de Riemann en argumentos enteros, tanto pares como impares. Además, se muestran ejemplos numéricos que evidencian la efectividad de dichos aproximantes.
La charla concluye mencionando algunos problemas abiertos en los cuales estamos trabajando actualmente.

Thursday, October 25 2012, 16:00. Room 2.2.D08

Title: Polynomial interpolation of analytic functions: There's life beyond extremality.

Abstract: We will review some of the best-known results on Taylor series and how they have been extended to the case of an analytic function interpolated along extremal tables of interpolation points. The divergence of the interpolants outside the critical equipotential curve, the overconvergence of the approximants provided the speed of convergence is geometric, and the Jentzsch-Szegö theorem will be considered. Then, we will discuss the main topic of the talk: Is it possible to prove such results for general tables of interpolation?

Thursday, October 18 2012, 16:00. Room 2.2.D08

Speaker: Jeffrey Geronimo, Georgia Institute of Technology.

Title: Orthogonal polynomials and wavelets.

Abstract: The refinement equation associated with multiresolution analysis when applied to the spline type multiwavelets provides some interesting combinatorial identities. The simplest case of this are the scaling functions of Alpert which are nothing but the Legendre polynomials.  I will review the theory of wavelets and then present some of these new identities. This is joint work with Professor Paco Marcellan.

Thursday, October 4 2012, 16:00. Room 2.2.D08

Title: Direct and inverse results on row sequences of simultaneous rational approximants.

Abstract: In this talk we investigate the approximation of vector functions by vector rational function that generalizes Padé approximants. We consider two types of approximants: the simultaneous Hermite-Pade approximants, which are constructed by mean of interpolation criterion and Fourier-Padé approximants based on Fourier series expansions in terms of a system of orthogonal polynomials. The results obtained in terms of generalize to the vector case results well known for the scalar case due to R. of Montessus of Ballore, A.A. Gonchar, S.P. Suetin, P.R. Graves-Morris, and E. B. Saff.

Thursday, September 27 2012, 16:00. Room 2.2.D08

Title: On Krall-type and Sobolev-type Orthogonal Polynomials.

Abstract: This talk will be focused on the so called Krall-type and Sobolev-type orthogonal polynomial sequences (OPS). In the first part, we analyze the zeros and some outer asymptotic properties of Krall-type OPS. We study several examples of perturbed measures supported in a finite or infinite interval of the real line. An electrostatic model for the zeros of Laguerre-Krall OPS is also considered. The second part of the talk is devoted to Sobolev-type OPS of unbounded support. In particular we obtain some asymptotics properties for discrete Laguerre-Sobolev-type and its interpretation as Matrix OPS.

Thursday, September 20 2012, 16:00. Room 2.2.D08