Charlas plenarias

Por orden alfabético:

María José Cantero (Universidad de Zaragoza, España)
Título: Darboux transformations for orthogonal polynomials on the real line and on the unit circle.
Abstract: Jacobi matrices connect Orthogonal Polynomials on the real line (OPRL), self-adjoint operators and certain integrable systems (Toda lattice), playing a different role in each of these fields.Jacobi matrices constitute the matrix representation of the recurrence relation for OPRL, provide the canonical representations of self-adjoint operators and are also essential in the formulation of the Toda lattice in terms of a Lax pair. The Darboux transformations for Jacobi matrices, which are equivalent to Christoffel modifications of the corresponding orthogonality measure, is one of the tools which best exploits this triple connection.
Each of the above topics has an analogue on the unit circle: Orthogonal Polynomials on the unit circle (OPUC) play a similar role to that of OPRL, and their recurrence relation is represented by the so called CMV matrices, the unitary version of Jacobi matrices. CMV matrices are also key in the Lax pair description of certain integrable systems. It has been recently discovered that Darboux transformations make sense in this new context, although some differences appear with respect to the Jacobi case.
In this talk we will present a survey of Darboux transformations for Jacobi and CMV matrices, relating them with OPRL, OPUC and integrable systems, and also highlighting their similarities and differences.

Alfredo Deaño (University of Kent, United Kingdom)
Título: Orthogonal polynomials and Painlevé equations.
Abstract: Painlevé equations are closely related to the theory of orthogonal polynomials, when classical weights on the real line or the complex plane are deformed in specific ways. In these cases, recurrence coefficients or Hankel/Toeplitz determinants can typically be written as special function solutions of some of the Painlevé equations. At the same time, these solutions arise in random matrix theory, for example in the calculation of gap probabilities for unitarily invariant ensembles such as GUE or LUE. In this talk we will illustrate these connections using Painlevé II and Painlevé IV as examples.

Diego Dominici (State University of New York at New Paltz, USA)
Título: Discrete semiclassical orthogonal polynomials.
Abstract: In this talk, we will present some past, present and future results on some families of polynomials orthogonal with respect to a functional that satisfies a Person equation for the difference operator. This is joint work with Francisco Jose Marcellán Español.

Ulises Fidalgo (Case Western Reserve University, USA)
Título: Non complete interpolatory quadrature rules.
Abstract: Non-complete interpolatory quadrature rules were introduced in the 90's by Bloom, Lubinsky, and Stahl. With this type of rules we can compute approximation of integrals with Riemann integrable functions, where the evaluation nodes distribution has an appreciable flexibility. In this context we find a wide family of convergent schemes of nodes. To this end we use some results on the orthogonal polynomial theory, such as: strong asymptotic behavior of orthogonal polynomial with respect to varying measures, and the Wendroff theorem.

Ana Foulquié Moreno (Universidad de Aveiro, Portugal)
Título: Riemann- Hilbert problem and Matrix Orthogonal Polynomials.
Abstract: In this talk we will analyze the formulation of the Riemann-Hilbert Problem in the theory of Orthogonal Polynomials and also in the theory of the Matrix Orthogonal Polynomials.

Judit Mínguez (Universidad de La Rioja, España)
Título: Fourier series of Gegenbauer-Sobolev polynomials.
Abstract: The study of orthogonal polynomials with respect to a Sobolev-type inner product
has attracted the interest of many researchers in the last years. The main target of our work is the study of the convergence of the Fourier series in terms of orthonormal polynomials with respect to an inner product as (1). Pollard studied the uniform boundedness of the partial sum operators in Lp for Gegenbauer and Jacobi polynomials, and as a consequence, the convergence of the Fourier series in Lp . In this talk, we will show a complete characterization of the uniform boundedness of the partial sums in some Sobolev cases in order to obtain convergence of Fourier
series. This is a joint work with O. Ciaurri.

Maria das Neves Rebocho (Universidade da Beira Interior, Portugal)
Título: On Laguerre-Hahn orthogonal polynomials on the real line.
Abstract: Laguerre-Hahn orthogonal polynomials on the real line were introduced by J. Dini in 1988, within the framework of theory of distributions and moment functionals, and, since then, studied by several authors within a vast list of problems, for instance, related to Hermite-Padé Approximation, measure modifications, and perturbations of orthogonal polynomials. A fundamental property of Laguerre Hahn orthogonal polynomials is the Riccati type differential equation for the corresponding Stieltjes function,
                                                      A·S’ = BS2 + CS + D ,    A ≠ 0 ,                                      (1)
where A, B, C, D are polynomials. From (1), the Laguerre-Hahn orthogonal polynomials are generalizations of semi-classical orthogonal polynomials, as the later ones appear upon the specification B ≡ 0. Furthermore, upon the bounds deg(A) ≤ 2, deg(C) = 1, we recover the classical orthogonal polynomials - the Hermite, Laguerre and Jacobi polynomials. In this talk I will survey some recent results on Laguerre-Hahn orthogonal polynomials, focusing on the difference-differential equations for the systems of polynomials and (recent) connections with Painlevé equations.

Javier Segura (Universidad de Cantabria, España)
Título: Numerical computation of classical Gaussian quadrature rules.
Abstract: The numerical computation of classical Gaussian quadrature rules has recently received a considerable attention, particularly regarding the computation of high degree rules. Most of the numerical methods are based on a combination of simple asymptotic methods for computing first estimations of the nodes, an iterative refinement of these nodes and the computation of the weights using well-known expressions derived from the Christoffel-Darboux formula. We show that direct asymptotic methods are enough for a fast and high accuracy computation of Gauss-Hermite, Gauss-Laguerre and Gauss-Jacobi quadratures, without recourse to iterative methods. And, on the other hand, we also show that it is possible to build rapidly convergent iterative methods which do not depend on asymptotics. These two separated techniques have a number advantages with respect to mixed methods: purely asymptotic methods are very fast and accurate for large degrees, while our iterative methods are valid without practical restrictions on the parameters and with arbitrary accuracy.

Jan Felipe Van Diejen (Universidad de Talca, Chile)
Título: Quadrature rules from finite orthogonality relations for Bernstein-Szegö polynomials.
Abstract: By gluing two families of Bernstein-Szegö polynomials, we construct  the eigenbasis of an associated finite-dimensional Jacobi matrix. This  gives rise to finite orthogonality relations for this composite  eigenbasis of Bernstein-Szegö polynomials. As an application, a number  of Gauss-type quadrature rules are derived for the exact integration  of rational functions with prescribed poles against the Chebyshev  weight functions. Based on work in collaboration with Erdal Emsiz  (Pontificia Universidad Católica de Chile).

Luis Verde Star (Universidad Autónoma Metropolitana, México)
Título: The extended q-Hahn class and the q-Askey scheme
Abstract: We will describe a class of q-orthogonal polynomial sequences that contains the q-Hahn class and all the families of q-orthogonal sequences in the q-Askey scheme, including the Askey-Wilson and the q-Racah polynomials, and many other sequences that are not listed in the q-Askey scheme. We will present explicit formulas for the recurrence coefficients of all the elements in the extended q-Hahn class using four parameters and show  how each family of sequences in the q-Askey scheme is obtained by direct substitution of the parameters with  appropriate values, without taking limits. In addition, we will discuss a general basic hypergeometric representation, in terms of the same four parameters, that yields the representation for each family in the q-Askey scheme by substitution of the parameters with appropriate values. 
The extended q-Hahn class was obtained using a matrix approach to study orthogonal polynomials and is closed under certain involutions that allow us to avoid taking limits to give particular values to the parameters in the general formulas.
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