## 7-9/02/2002

### Departamento de Matemática da Faculdade de Ciências e Tecnologia da Universidade de Coimbra

With the support of Centro de Matemática da Universidade de Coimbra and INTAS project ref. 00-272 .

### 1Introduction

#### Scientific Committee:

Walter Van Assche (Katholieke Universiteit Leuven), Alexander Ivanovich Aptekarev (Keldysh Institute of Applied Mathematics) and Amílcar Branquinho (Coimbra University).

#### Organizing Committee:

Ana Foulquié (Aveiro University) and Amílcar Branquinho (Coimbra University).

#### Participants:

Students and researchers of Portugal and members of the INTAS project ref. 00-272.

#### Goals:

The INTAS workshops is a place where all the teams of the project will meet and where they will be able to present the obtained results and to make plans to continue the research.

#### Place:

The meeting take place in room Anastácio da Cunha of Mathematical Department of  Coimbra University.

#### Cultural Program:

February 7:  visit to the University of Coimbra
February 8:  visit to the Mathematical Department library; Conference Dinner

#### Accommodation:

The participants stay in Residenciais Alentejana, Antunes and Botânico.

### 2Work Programme

 7-02-2002 8-02-2002 9-02-2002 9h30m-10h30m × Kaliaguine × 10h30m-11h × Castro/Durán meeting 11h-11h30m opening session meeting 11h30m-12h Stahl Marcellán closing session 12h-12h30m --×-- Martinez Finkelshtein × 12h30m-15h lunch/visit lunch/visit lunch 15h-16h Van Assche Aptekarev × 16h-16h30m Beckerman Van Iseghem × 16h30m-17h Arvesu Kuijlaars × 17h-17h30m Buyarov López Lagomasino × 17h30m-19h workshop seminar workshop seminar × 19h × Dinner ×

### 3Talks

Author: Herbert Stahl

Title: Hermite-Padé Polynomials and Algebraic Approximants to the Exponential Function

Abstract: Results about Hermite-Padé polynomials and algebraic approximants to the exponential function will be reviewed. Special emphasis is given to the asymptotic distribution of the zeros of Hermite-Padé polynomials and the definition of a Riemann surface which is pivotal in the description of the asymptotic behavior of the polynomials, its zeros, and the error function of the algebraic approximants associated with the exponential function. We shall report about new results, which are part of the coordinated activities of the INTAS project.

Author: Alexander Aptekarev

Title:  Strong asymptotics for polynomials orthogonal with respect to a complex varying weight and applications to the best rational approximation

Abstract: We discuss application of methods based on boundary value problems for analytic functions (Riemann-Hilbert problem) to the asymptotics of complex orthogonal polynomials. Statements of two theorems in this direction will be considered in details. These theorems are the key tool for obtaining the sharp constants for the rate of approximation by rational functions of some general class of analytic functions, which includes exp, log and sqrt.

Author: Andrei Martinez Finkelshtein

Title: Electrostatics for the zeros of classical polynomials revisited

Abstract:  This is a preliminary report on the progress of the electrostatic interpretation of the zeros of some classical polynomials, which avoids an a priori restriction on the location of the zeros. It is shown that the zeros solve certain min-max problem for the logarithmic energy, extending (in some since) to the discrete setting the well-known symmetry of the compact of min capacity, described by H. Stahl.

Only the simplest case of Jacobi polynomials with parameters >-1 will be considered; although the proof still presents some gaps, its present state makes it worth to be discussed.

Author:

Title: The relativistic Toda lattice

Abstract: The relativistic Toda lattice is a generalization of the ordinary Toda lattice introduced by S. Ruijsenaars in 1990.  It is well-known that the finite, non-periodic Toda lattice is related to  spectral theory of tridiagonal matrices and orthogonal polynomials.  I will discuss an analogous relation in the relativistic case with spectral theory for a pair of bidiagonal matrices and  Laurent orthogonal polynomials.

This is work in progress with Jonathan Coussement and Walter Van Assche (both at K.U.Leuven).

Author: Walter Van Assche

Abstract: We investigate the asymptotic behavior of the polynomials $p,q,r$ (with degrees respectively $n_1,n_2,n_3$)  in type I Hermite-Pad\'e approximation to the exponential function, i.e.,  $p(z)e^{-z} +q(z) + r(z) e^z = \mathcal{O}(z^{n_1+n_2+n_3+2})$ as $z \to 0$. We use a Riemann-Hilbert approach for $3\times 3$ matrix valued functions. An important role is played by a three-sheeted Riemann surface and certain rational functions defined on it. We show how to transform the Riemann-Hilbert problem and how a deformation of the contours allows us to obtain asymptotics for the polynomials $p,q,r$. As a result, we obtain results on the behavior of the zeros of these polynomials.

Title: Some open problems in the theory of continued fraction

Abstract: We will discuss an approach to some open problem of Gonchar about discs of meromorphicity for continued C-fraction with randomly chosen coefficients of value +1 and -1.

Author: Jeannette Van Iseghem

Title: Vector QD algorithm and vector continued fractions

Abstract: Continued fractions and QD algorithm: scalar, vector and matrix cases.

Author: Jorge Arvesu

Title: Discrete Multiple Orthogonal Polynomials

Abstract: The talk deals with Discrete Multiple Orthogonal Polynomials. Special attention will paid to the q-Hermite-Padé polynomials. Some algebraic and analytical properties of the q-multiple orthogonal polynomials are shown. Finally, one example of q-Hermite-Padé polynomial is given.

Author: Francisco Marcellán

Title: Freud-Sobolev Orthogonal Polynomials. Asymptotic properties

Abstract: We introduce orthogonal polynomials with respecto to a  Sobolev inner product involving the first derivatives where  the measure is associated with the weight function exp -x4 coinsidered among others by G. Freud and P. Nevai. In particular we study some algebraic and analytic properties  of such polynomials. A Plancherel-Rotach asymptotic formula is obtained.

Author: Guillermo López Lagomasino

Title: Stieltjes polynomials with respect to varying weigths

Abstract:

Title: Boundedness properties for Sobolev inner products

Abstract: The work deals with Sobolev orthogonal polynomials. Sobolev orthogonality is written in terms of matrix orthogonality, involving differential operators. Particularly, it is explored the connection bettween some boudedness properties for the support of the measures appearing in the Sobolev inner product and the set o  zeros of the orthogonal polynomials, completing previous results in this area.

Author: Bernhard Beckermann

Title: The (discrete) Zolotarev Problem and its applications to best rational approximation and to eigenvalue estimates for Hankel matrices

Abstract: Recently, some approximately sharp upper bounds were given for the ratio of the smallest over the largest eigenvalue of any positive definite Hankel matrix. In the present talk we present similar results for the ratio of two eigenvalues of a positive definite Hankel matrix. This enables us to derive lower bounds for the condition number of any (not necessarily positive definite) Hankel matrix in terms of its signature.

A main tool in our investigations is some extremal problem in logarithmic potential theory, namely a condenser with a maximum charge constrain.

Author:

Title: A counter example to the Padé conjecture

Abstract: Recently D. Lubinsky has obtained a counter-example for long standing conjecture of Backer-Gammel-Wills (or the so called Padé-conjecture). His counter-example related to the so called meromorphic case of the conjecture. We shall speak about another new example related to the holomorphic´´ case of the conjecture. Thus a function holomorphic in a circle such that any subsequence of its diagonal Padé approximation does not converge locally uniformly inside the circle, will be presented.

Author: Valeri Kaliaguine

Title: On compact perturbation of band operators

Abstract: We will discuss about inversion of Weyl criterion of compact perturbation of operators. Thus, we consider a problem whether adding to the spectral measure of a Jacobi or band operator discrete masses, we get or not a compact perturbation of operator.

### 4List of Participants

Alexander Ivanovich Aptekarev - Russian Academy of Sciences, Keldysh Institute of Applied Mathematics, Miusskaya Square 4, 125047 Moscow, RUSSIA
e.mail: aptekaa@rfbr.ru
Accomodation: Residencial Antunes

Alexandra Nascimento - Instituto Politécnico de Leiria, Leiria, PORTUGAL
e.mail: xana_nascimento@hotmail.com
Accomodation: Residencial Antunes

Amílcar Branquinho - Departamento de Matemática da Universidade de Coimbra, 3000 Coimbra, PORTUGAL
e.mail: ajplb@mat.uc.pt

Ana Filipa Soares Loureiro - ISEC, Coimbra, PORTUGAL
e.mail: anafsl@isec.pt

Ana Foulquié Moreno - Departamento de Matemática da Universidade de Aveiro, 3810-193 Aveiro, PORTUGAL
e.mail: foulquie@mat.ua.pt

Ana Isabel Mendes - Instituto Politécnico de Leiria, Leiria, PORTUGAL
e.mail: aimendes@yahoo.com

Ana Margarida Santos - Departamento de Matemática da Universidade de Aveiro, 3810-193 Aveiro, PORTUGAL
e.mail: @

Anabela Monteiro Paiva - Departamento de Matemática da Universidade da Beira Interior, 6200 Covilhã, PORTUGAL
e.mail: apaiva@noe.ubi.pt
Accomodation: Residencial Antunes

André Draux -  INSA de ROUEN, Rouen, FRANCE
e.mail: draux@lmi.insa-rouen.fr
Accomodation: Residencial Botânico

e.mail: andrei@ual.es
Accomodation: Residencial Botânico

Antonio José Durán Gardeño - Universidad de Sevilla, SPAIN
e.mail: duran@us.es
Accomodation: Residencial Antunes

Arno B. J. Kuijlaars - Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven (Heverlee), BELGIUM
e.mail: Arno.KUIJLAARS@wis.kuleuven.ac.be
Accomodation: Residencial Alentejana

Bernhard Beckermann - Université de Lille 1, Laboratoire d'Analyse Numérique et d'Optimisation, Cité Scientifique, F-59655 Villeneuve d'Ascq, FRANCE
e.mail: bbecker@ano.univ-lille1.fr
Accomodation: Residencial Antunes

Claude Brezinski - Université de Lille 1, Laboratoire d'Analyse Numérique et d'Optimisation, Cité Scientifique, F-59655 Villeneuve d'Ascq, FRANCE
e.mail: Claude.Brezinski@univ-lille1.fr
Accomodation: Residencial Botânico

Christian Berg - K\obenhavns Universitet, Copenhagen, DENMARK
e.mail: BERG@math.ku.dk
Accomodation: Residencial Antunes

Elisabete Sousa Almeida - Instituto Politecnico de Viseu, Viseu, PORTUGAL
e.mail: betty@mat.estv.ipv.pt
Accomodation: Residencial Antunes

e.mail: pacomarc@ing.uc3m.es
Accomodation: Residencial Alentejana

e.mail: lago@math.uc3m.es
Accomodation: Residencial Alentejana

Helmuth R. Malonek - Departamento de Matemática da Universidade de Aveiro, 3810-193 Aveiro, PORTUGAL
e.mail: hrmalon@mat.ua.pt

Herbert Stahl - Technische Fachhochschule Berlin, Fachbereich II, Luxemburgerstrasse 10, D-13353 Berlin, GERMANY
e.mail: stahl@tfh-berlin.de
Accomodation: Residencial Botânico

Jaime Carvalho e Silva - Departamento de Matemática da Universidade de Coimbra, 3000 Coimbra, PORTUGAL
e.mail: jaimecs@mat.uc.pt

Jeannette Van Iseghem - Université de Lille 1, Laboratoire d'Analyse Numérique et d'Optimisation, Cité Scientifique, F-59655 Villeneuve d'Ascq, FRANCE
e.mail: jvaniseg@ano.univ-lille1.fr
Accomodation: Residencial Antunes

e.mail: jarvesu@math.uc3m.es
Accomodation: Residencial Antunes

e.mail: jsanchez@math.uc3m.es
Accomodation: Residencial Antunes

José Carlos Petronilho - Departamento de Matemática da Universidade de Coimbra, 3000 Coimbra, PORTUGAL
e.mail: josep@mat.uc.pt

Luis Cotrim - Instituto Politécnico de Leiria, Leiria, PORTUGAL
e.mail: lmsc@estg.iplei.pt
Accomodation: Residencial Antunes

Luis Daniel Moura Abreu - Departamento  de Matemática da Universidade de Coimbra, 3000 Coimbra, PORTUGAL
e.mail: daniel@mat.uc.pt

Maria das Neves Vieiro Rebocho - Departamento de Matemática da Universidade da Beira Interior, 6200 Covilhã, PORTUGAL
e.mail: mneves@noe.ubi.pt
Accomodation: Residencial Antunes

Maria Francisca Matos Cabo - Departamento de Matemática da Universidade de Coimbra, 3000 Coimbra, PORTUGAL
e.mail: mfmc@mat.uc.pt

Mário António Grande Abrantes - Instituto Politécnico de Bragança, Bragança, Portugal
e.mail: mar@ipb.pt

Mirta Castro Smirnova - Moscow State University (Moscow - RUSSIA) and Universidad de Sevilla (Sevilla - SPAIN)
e.mail: mirta@cica.es
Accomodation: Residencial Antunes

e.mail: ufidalgo@math.uc3m.es
Accomodation: Residencial Antunes

Valeri Aleksandrovich Kalyagin (Kaliaguine) - Nizhny Novgorod State Technical University, Department of Applied Mathematics, Minina Street 24, 603600 Nizhny Novgorod, RUSSIA
e.mail: kalia@waise.nntu.sci-nnov.ru
Accomodation: Residencial Antunes

Victor I. Buslaev - Steklov Mathematical Institute, Moscow, RUSSIA
e.mail: buslaev@mi.ras.ru
Accomodation: Residencial Antunes