|
|
|
|
|
|
|
|
|
|
|
Introdução
dos
interessados
na
Teoria
dos
Polinómios
Ortogonais e Aplicações,
bem
como
incentivar
o
debate
de
problemas
nesta área. |
|
|
-
Teoria
Geral
dos
Polinómios Ortogonais
-
Teoria
das
Equações
em
Diferenças.
Teorema
de
Poincaré
e Aplicações
-
Problema
de Momentos
-
Teoria
do Potencial
-
Teoria
Analítica dos Números
|
|
|
|
Autor: Mário António Grande Abrantes |
|
Titulo:
Polinómios Ortogonais
de Uma e Duas Variáveis |
|
Defesa: Coimbra, Julho de 2001 |
|
Autor: Maria das Neves Vieiro Rebocho |
|
Titulo:
Teoria de Momentos e
Polinómios Ortogonais |
|
Defesa: Coimbra, Julho de 2001 |
|
Autor: Ana Isabel Gonçalves Mendes |
|
Titulo: Geração Semi-Clássica de Famílias de
Polinómios Ortogonais |
|
Defesa: Coimbra, Março de 2002 |
|
Autor: Ana Margarida Santos |
|
Titulo: Polinómios Ortogonais Matriciais: Caso Clássico |
|
Defesa: Aveiro, Julho de 2002 |
|
Autor: Anabela Monteiro
Paiva |
|
Titulo: Polinómios
Ortogonais Tipo Pollaczek |
|
Defesa: Coimbra, Dezembro de
2002 |
|
Autor: Elisabete Sousa Almeida |
|
Titulo: Uma Abordagem à Teoria de Funções |
|
Defesa: Coimbra, Junho de 2004 |
|
Autor: Victor Luís Pereira de Sousa |
|
Titulo: Problemas de Riemann-Hilbert na Teoria dos
Polinómios Ortogonais |
|
Defesa: Aveiro, Junho de 2004 |
|
Autor: Alexandra Nascimento |
|
Titulo: Teoria Analítica dos Números |
|
Defesa: Coimbra,
Setembro de 2005 |
|
|
|
Autores: A. Branquinho, A.
Foulquié Moreno and F. Marcellán |
|
Titulo: Asymptotic behavior of
Sobolev type orthogonal polynomials on a recti able Jordan Curve or Arc |
|
Referência: Constructive Approximation,
18 (2002) 161-182 |
|
Autores: A.
Aptekarev, A. Branquinho and W. Van Assche |
|
Titulo: Classical multiple
orthogonal polynomials |
|
Referência:
Trans. Amer.
Math. Soc. vol. 335, N. 10, pp. 3887-3914 |
|
Autores: A.
Aptekarev and A. Branquinho |
|
Titulo: Padé approximants and complex high order Toda lattices |
|
Referência: Journal of Computation and Applied Mathematics
2004 |
|
Autores: A. Branquinho, J. Bustamante, A.
Foulquié Moreno and G. López |
|
Titulo: Normal indices in Nikishin Systems |
|
Referência: Journal of Approximation Theory 2004 |
|
Autores: A. Branquinho, A. Foulquié Moreno, F. Marcellán, and K. Pan |
|
Titulo: Coherent Pairs in Jordan Arcs and Curves |
|
Referência: submitted |
|
Projectos de Doutoramento em curso |
voltar ao início |
|
Tema de Trabalho: Famílias de Polinómios Ortogonais |
|
Destinatário: Ana Isabel Gonçalves Mendes |
|
Directores: Francisco Marcellán e Amílcar Branquinho |
|
Resumo: |
O objectivo principal do plano de trabalho é o de analisar
alguns métodos construtivos para sucessões de polinómios ortogonais,
relativamente a uma funções de Stieltjes ou a uma funcional linear contínua.
Num sentido amplo podemos dizer que o nosso principal interesse centra-se no
estudo das famílias de polinómios
ortogonais semi-clássicas e de Laguerre-Hahn.
Vamos começar por estabelecer processos de geração de famílias de polinómios
ortogonais e posteriormente analisar a estabilidade destas famílias
relativamente a modificações, na função de Stieltjes associada, e nos
coeficientes da relação de recorrência a três termos que estas famílias
verificam. |
|
Tema de Trabalho: Propriedades
analíticas de sistemas de polinómios ortogonais relativamente a formas diferenciais |
|
Destinatário: Maria das Neves Vieiro Rebocho |
|
Director: Amílcar Branquinho |
|
Resumo: |
O objectivo principal do plano de trabalho é o estudo das
propriedades analíticas de sistemas de polinómios ortogonais gerados por
relações de ortogonalidade não usuais, que surgem em diversas aplicações da
análise, do cálculo numérico, da física teórica, das equações diferenciais e da
teoria da aproximação.
Num sentido amplo podemos dizer que o nosso foco de interesse são os sistemas de
polinómios, ortogonais relativamente a uma forma diferencial. Estes sistemas
incluem como casos particulares os chamados de tipo Sobolev, os de
ortogonalidade incompleta e os de ortogonalidade múltipla.
Por um lado pretendemos estabelecer os fundamentos de uma teoria geral de tais
sistemas e por outro, aproveitando as
características específicas de alguns subsistemas destes produtos, avançar no
estudo analítico dos mesmos. |
|
Tema de Trabalho: Ortogonalidade Múltipla |
|
Destinatário: Luis Manuel da Silva Cotrim |
|
Director: Amílcar Branquinho |
|
Resumo: |
|
O objectivo principal do plano de doutoramento é o
estudo de funções especiais, dando particular atenção, ao estudo de
sucessões de polinómios ortogonais múltiplos de variável discreta e
contínua. |
|
Tema de Trabalho: Polinómios Ortogonais Tipo
Askey-Wilson |
|
Destinatário: Anabela Monteiro Paiva |
|
Director: Guillermo López e Amílcar Branquinho |
|
Resumo: |
|
O objectivo principal deste plano de doutoramento é o
estudo de problemas inversos diferenciais na teoria dos polinómios
ortogonais, i.e. obter a medida associada a uma dada sucessão de polinómios
ortogonais, que verificam um determinado operador diferencial e/ou em
diferenças. |
|
Projectos
internacionais |
voltar ao início |
|
Titulo: Rational approximation of analytic
functions and its applications to the spectral theory of difference
operators, non-linear dynamical systems, special functions, and number
theory |
|
Referência:
INTAS-00-272 |
|
Titulo:
Orthogonal Polynomials
and Special Functions |
|
Referência:
29242-IC-1-2001-PT-ERASMUS-IP-13 |
|
Titulo: Network on Constructive Complex
Approximation (NeCCA) |
|
Referência: INTAS
Research Network |
|
Cursos de pós-graduação |
voltar ao início |
|
Alexander Aptekarev |
Two Courses on
Orthogonal Polynomials and Special Functions |
|
Title:
Scalar and Matrix Riemann Problem approach to the
asymptotics of polynomials orthogonal with respect to a
complex weight.
|
|
Abstract |
|
1.
Cauchy integral theorem and solution of the jump problem.
|
|
2. Short introduction to Riemann surfaces, meromorphic differentials. |
|
3. Cauchy Integral theorem on the Riemann surfaces and solution of Boumdary
Value Problems. |
|
4. Scalar Riemann problem approach of J.Nuttall to the asymptotics of complex
orthogonal polynomials. |
|
5. Matrix Riemann problem approach of P.Deift to the asymptotics of complex
orthogonal polynomials.
|
|
Guillermo López |
Two Courses on
Orthogonal Polynomials and Special Functions |
|
Title: Potential Theory |
|
Abstract |
|
The course will cover the main properties of the logaritmic potential up to
the proof of the existence of the equilibrium measure. Applications of this
theory to aymptotics of orthogonal polynomials and convergence of interpolatory
processes in rational approximation will be seen. |
|
Francisco Marcellán |
6th European
Intensive Course on Complex Analysis |
|
Title:
Orthogonal Rational Functions |
|
Abstract |
|
We will present the state of the art in the subject which
constitutes a new and interesting subject of research with many applications
in Linear Prediction, network synthesis and control theory: |
| 1.
The fundamental spaces
|
| 2.
Kernel functions, recurrence and second kind functions
|
| 3. Para-orthogonality and Quadrature
Density of rational functions
|
| 4. Convergence
|
|
5. The boundary case
|
|
6.
Applications
|
|
Francisco Marcellán |
7th European
Intensive Course on Complex Analysis |
|
Title:
Operator theory and moment problems |
|
Abstract |
|
The aim of this course is to introduce the state-of-the art of moment problems from the perspective of the theory of finite
difference operators in the scalar and in the matrix case, respectively. One the advantages of this approach is that the
Nevanlinna functions appear as elements of a transfer matrix and the Padé approximants are the resolvents of a finite
matrix approximation to a infinite Jacobi matrix.
|
|
Francisco Marcellán |
8th European
Intensive Course on Complex Analysis |
|
Title:
Approximation theory and weighted
Sobolev spaces |
|
Abstract |
| 1.
Basic results concerning weights.
|
| 2. Weighted Sobolev spaces. Isoperimetric inequalities. Sobolev type
inequalities.
|
| 3. Potentials and capacities. Meyer's Theory for Lp capacities. Some examples:
Bessel, Riesz and Haussdorff capacities.
|
| 4. Applications of Potential Theory to Sobolev spaces. Poincaré type
inequalities.
|
| 5.
Polynomial Approximation in Sobolev spaces. |
|
Guillermo López |
8th European
Intensive Course on Complex Analysis |
|
Title:
Geometric theory of functions of a
complex variable |
|
Abstract |
|
The course aims to cover basic questions related with the theory
of univalent conformal mappings of simply connected and multiply connected
domains, conformal mapping of multiply connected domains onto a disk (generalization
of Riemann's Mapping Theorem), applications of conformal mappings to the
study of interior and boundary properties of analytic functions, and general
questions of geometric nature dealing with analytic functions. |
|
Walter Van Assche |
9th European
Intensive Course on Complex Analysis |
|
Title:
Hermite-Padé
Approximation and Multiple Orthogonal Polynomials |
|
Abstract |
|
Hermite-Padé approximation is simultaneous
rational approximation of a vector (f1, … , fr)
of r functions with interpolation conditions at a given point in
the complex plane. In this course the following aspects will be covered:
- Padé approximation of a function. Relation with orthogonal polynomials.
- Hermite-Padé approximation of r functions. Type I
approximation and type II approximation.
- Orthogonality conditions for Hermite-Padé approximation: multiple
orthogonal polynomials.
- Special systems: Angelesco system, algebraic Chebyshev systems (AT
systems), Nikishin systems.
- Examples of multiple orthogonal polynomials.
- Analytic properties of Hermite-Padé approximation for Angelesco
systems and Nikishin systems.
- Riemann-Hilbert approach for the asymptotic analysis of Hermite-Padé
approximation.
- Application: simultaneous Gauss quadrature.
- Applications in number theory: transcendence of e,
irrationality of ζ(3).
|
|
Walter Van Assche |
10th European
Intensive Course on Complex Analysis |
|
Title: Riemann-Hilbert problem for
Orthogonal Polynomials |
|
Abstract |
|
A Riemann-Hilbert problem consists of finding an analytic function in the
complex plane minus a collection of oriented contours, for which the
boundary values on the contours (from both sides of the contours) are given.
Typically the solution will involve the Cauchy transform of a function w
on the contours, where w describes the boundary values on the
contours. The basic idea of the Riemann-Hilbert approach to orthogonal
polynomials is to characterize orthogonal polynomials corresponding to a
weight function w on the real line via a boundary value problem for
matrix valued analytic functions. This Riemann-Hilbert problem was first
formulated by Fokas, Its and Kitaev in 1992 and we will formulate the
appropriate Riemann-Hilbert problem for orthogonal polynomials on ]- ,+[,
[0,+[ and [-1,1].
We show that this Riemann-Hilbert problem enables us to find the three-term
recurrence relation and the differential equation for Hermite polynomials,
Laguerre polynomials, and Jacobi polynomials. One of the main advantages of
this Riemann-Hilbert approach is that it allows to obtain strong uniform
asymptotics which is valid in the whole complex plane. The idea is to
transform the initial Riemann-Hilbert problem in a few steps to another
equivalent Riemann-Hilbert problem on a deformed set of contours with
simpler jumps on the contours. This technique is called the steepest descent
method for Riemann-Hilbert problems and was developed by Deift and Zhou in
1993. If time permits we will show how it works to obtain the strong
asymptotics for Laguerre polynomials.
|
|
Andrei Martínez Finkelshtein |
10th European
Intensive Course on Complex Analysis |
|
Title: Potential Theory |
|
Abstract |
|
The course will cover the main properties of the logarithmic potential up
to the proof of the existence of the equilibrium measure. Applications.
|
|
Escolas e Congressos |
voltar ao início |
|
Summer
School on Orthogonal Polynomials and Spcial Functions:
Recent Trends in Computation and Applications,
Madrid, July 2004 |
|
10
Intensive Course on Complex Analysis |
|
7
Workshop on Complex Analysis |
|
Summer School on Orthogonal Polynomials and
Special Functions:Approximation and Iteration, Coimbra, July
2003 |
|
9th Intensive Course on Complex Analysis, Coimbra, March 2003 |
|
6 Workshop on Complex Analysis,
Aveiro, March 2003 |
|
8th
Intensive
Course on Complex Analysis, Coimbra,
March 2002 |
|
5 Workshop on Complex Analysis,
Coimbra, March 2002 |
|
Rational Approximation and its Applications,
Coimbra, February 2002 |
|
Two Courses on Complex Analysis,
Coimbra, July 2001 |
|
Computational Methods and Function Theory 2001,
Aveiro, June 2001 |
|
7th
Intensive
Course on Complex Analysis, Coimbra,
March 2001 |
|
4 Workshop on Complex Analysis,
Coimbra, March 2001 |
|
6th
Intensive
Course on Complex Analysis, Coimbra,
March 2000 |
|
Reunião Ibérica de Polinómios Ortogonais,
Coimbra, November 1999
|