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Weekly Seminar
 WS Season 2015/2016

# Seminar on Orthogonality, Approximation Theory and Applications

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## Past seminars

Thursday, May 26 2016, 16:00. Room 2.2.D08

Speaker: Walter Van Assche (KU Leuven, Belgium)

Title: Riemann-Hilbert analysis for a Nikishin system.

Abstract: In this seminar I will outline a method to obtain the asymptotic behavior of multiple orthogonal polynomial (and Hermite-Pad\’e approximants) for a Nikishin system of order 2.  The method is based on a Riemann-Hilbert problem for 3x3 matrices, which is an extension  of the Riemann-Hilbert problem for orthogonal polynomials introduced by Fokas, Its an Kitaev.  The Riemann-Hilbert analysis consists of a number of transformations of the original Riemann-Hilbert problem to one for which the asymptotic behavior is known. Each transformation uses important  information: the Nikishin construction, the equilibrium problem, the geometry of the problem, and the weight functions on the intervals. This is ongoing work with Guillermo López Lagomasino.

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Thursday, May 19 2016, 16:00. Room 2.2.D08

Speaker: Luis Verde-Star (Universidad Autónoma Metropolitana, México)

Title: Operadores en diferencias del tipo de Bochner que tienen polinomios ortogonales como autofunciones.

Abstract: Usando un enfoque matricial encontramos una familia de operadores en diferencias que generalizan a los operadores de Bochner de q-diferencias de orden dos con coeficientes polinomiales. El operador clásico de q-diferencias y el de (1/q)-diferencias son reemplazados por un par de operadores U y V que dependen de dos parámetros (r, t). Se cumple que para cada (r,t) y cada par de polinomios h(x), de grado uno, y g(x), de grado menor que o igual a dos, existe una sucesión de polinomios ortogonales p_k(x) que satisface la ecuación de valores propios (h(x) V + g(x) V U) p_k(x)=w_k  p_k(x)  para una sucesión de números w_k que depende del operador. Encontramos fórmulas explícitas para los coeficientes de los polinomios y para los coeficientes de la relación de recurrencia de tres términos.

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Thursday, May 12 2016, 16:00. Room 2.2.D08

Speaker: Nattapong Bosuwan (Mahidol University, Bangkok)

Title: Determining singularities using row sequences of linear Padé-orthogonal approximation and its extension

Abstract: We study the relation of the convergence of poles of row sequences of both linear and nonlinear Padé-orthogonal approximants (Padé approximants of orthogonal expansions) and the singularities of the approximated function. We  prove both direct and inverse results for these row sequences. Thereby, we obtain analogues of the theorems of Montessus de Ballore, Gonchar, and Fabry. If the time allows, then we will discuss our recent results on convergence of orthogonal Hermite-Padé.

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Thursday, April 21 2016, 16:00. Room 2.2.D08

Speaker: Jesús María Sanz-Serna (Universidad Carlos III de Madrid)

Title: Gauss' Gaussian Quadrature

Abstract: Gauss introduced what we now call Gaussian quadrature in 1814. While modern presentations follow Jacobi and use orthogonal polynomials, Gauss original approach was quite different. In an impressive display of virtuosity, he employed generating functions/Z-transforms, continued fractions, Pade approximation, hypergeometric functions, and other tools. As a modern numerical analyst would do, he concluded his work by providing, with great accuracy, the required nodes and weights for the formulas with 1, ..., 7 nodes and reporting in detail a numerical experiment. My talk will be a guided, critical reading of his Methodus nova integralium valores per approximationem inveniendi, published in MDCCCXV.

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Thursday, April 7 2016, 16:00. Room 2.2.D08

Speaker: Edmundo Huertas (Universidad de Alcalá)

Title: Isometries between martingale spaces and orthogonal polynomials

Abstract: Sets of orthogonal martingales are important because they can be used as stochastic integrators in a kind of chaotic representation property. In this talk, we revisited the problem studied by W. Schoutens in [1], investigating how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization procedure of a sequence of martingales related to a certain Lévy process, called Teugels Martingales. Since the Uvarov transformation depends by a c<0, we are studying how to provide infinite sets of strongly orthogonal martingales, each one for every c in (-\infty ,0). In a similar fashion as in [2] (see also [3]), our aim is to introduce a suitable isometry between the space of polynomials and the space of linear combinations of Teugels martingales as well as the general orthogonalization procedure.

[1] D. Nualart and W. Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Process. Appl. 90, (2000) 109--122.

[2] W. Schoutens, An application in stochastics of the Laguerre-type polynomials, J. Comput. Appl. Math. 133, (1-2) (2001), 593--600.

[3] W. Schoutens, Stochastic processes and orthogonal polynomials, in Lecture Notes in Statist. 146, Springer-Verlag, New York, (2000).

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Thursday, March 31 2016, 16:00. Room 2.2.D08

Speaker: José Manuel Rodríguez (Universidad Carlos III de Madrid)

Title: Stability of the injectivity radius under quasi-isometries and applications to isoperimetric inequalities.

Abstract: Kanai proved the stability under quasi-isometries of numerous global properties (including isoperimetric inequalities) between Riemannian manifolds of bounded geometry. Even though quasi-isometries highly distort local properties, recently it was shown that the injectivity radius is preserved (in some appropriate sense) under these maps between zero genus Riemann surfaces. In the present work, results along these lines are obtained even for infinite genus.  As a consequence, the stability of the isoperimetric inequality in this context  (without the hypothesis of bounded geometry) is also obtained.

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Thursday, March 17 2016, 16:00. Room 2.2.D08

Speaker: Lino Gustavo Garza Gaona (Universidad Carlos III de Madrid)

Title: A matrix approach for the classical, semiclassical and coherent orthogonal polynomials

Abstract: I will present a recent matrix characterization for the classical orthogonal polynomials. We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals.

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Thursday, March 10 2016, 16:00. Room 2.2.D08

Speaker: Daniel A. Rivero Castillo (Universidad Carlos III de Madrid)

Title: Ceros de primitivas de polinomios ortogonales clásicos

Abstract: En esta charla se presentarán resultados acerca de la localización de ceros y comportamiento asintótico de primitivas de polinomios ortogonales standard con respecto a una medida positiva de Borel concentrada en la recta real. Como casos particulares se estudian las primitivas de polinomios ortogonales clásicos, es decir los casos Jacobi, Laguerre y Hermite. Algunos ejemplos ilustrativos serán mostrados.

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Thursday, March 3 2016, 16:00. Room 2.2.D08

Speaker: Amauris de la Cruz (Universidad Carlos III de Madrid)

Title: Hyperbolicity of direct products of graphs

Abstract: If X is a geodesic metric space and x1, x2, x3  in X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is delta -hyperbolic (in the Gromov sense) if any side of T is contained in a \delta -neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by delta (X) the sharp hyperbolicity constant of X, i.e., delta (X) = inf{ delta >= 0 : X is delta-hyperbolic}. Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product G1 G2 is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we fi nd characterizations which are not so simple. Furthermore, we obtain good bounds for the hyperbolicity constant of the direct product of some important graphs.

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Thursday, February 25 2016, 16:00. Room 2.2.D08

Speaker: Víctor Elvira (Dpto. Teoría de la Señal y Comunicaciones, Universidad Carlos III de Madrid)

Title: Generalized Multiple Importance Sampling

Abstract: Importance Sampling methods are broadly used to approximate posterior distributions or some of their moments. In its standard approach, samples are drawn from a single proposal distribution and weighted properly. However, since the performance depends on the mismatch between the targeted and the proposal distributions, several proposal densities are often employed for the generation of samples. Under this Multiple Importance Sampling (MIS) scenario, many works have addressed the selection or adaptation of the proposal distributions, interpreting the sampling and the weighting steps in different ways. In this work, we propose a novel general framework for sampling and weighting procedures when more than one proposal is available. The most relevant MIS schemes in the literature are encompassed within the new framework, and, moreover novel valid schemes appear naturally. All the MIS schemes are compared and ranked in terms of the variance of the associated estimators. Finally, we provide  illustrative examples which reveal that, even with a good choice of the proposal densities, a careful interpretation of the sampling and weighting procedures can make a significant difference in the performance of the method. Preprint available at: http://arxiv.org/abs/1511.03095

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Thursday, February 11 2016, 16:00. Room 2.2.D08

Speaker: Javier Segura (Universidad de Cantabria)

Title: Evaluación eficiente de cuadraturas Gaussianas de orden alto

Abstract: La computación de cuadraturas Gaussianas es un problema clásico que se puede abordar mediante el conocido algoritmo de Golub-Welsch. Sin embargo, este método, basado en la  diagonalización de una matriz de tamaño N (siendo N el número de nodos), no es eficiente cuando el número de nodos es grande. En su lugar, la evaluación de los ceros (nodos de la cuadratura gaussiana) escogiendo
valores iniciales suficientemente precisos y aplicando algún método iterativo para refinar estos valores es más efectiva en estos casos. Los tres ingredientes básicos de los métodos iterativos de cálculo de cuadraturas gaussianas son, en primer lugar, la estimación de los valores iniciales para iniciar el proceso, en segundo la construcción de un algoritmo de evaluación de los polinomios ortogonales implicados y finalmente el refinamiento iterativo de los valores obtenidos en el primer paso mediante un método iterativo (que utiliza los valores del polinomio evaluados mediante el algoritmo del segundo paso).

En este charla describimos el estado actual de la cuestión y cómo es crucial la elección de un método iterativo adecuado al problema. En particular, describiremos la utilización de métodos iterativos de cuarto orden de tipo Sturm (basados en el teorema de comparación) que permiten calcular nodos de forma eficiente y global, hasta el punto de que no es necesaria una estimación previa. En cualquier caso, esas estimaciones pueden mejorar la eficiencia del algoritmo y discutiremos también de forma breve cómo utilizar información asintótica para evaluar eficientemente tanto las estimaciones iniciales como el valor de los polinomios y su derivada.

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Thursday, January 21 2016, 16:00. Room 2.2.D08

Speaker: Ramon Orive (Universidad de La Laguna)

Title: Sobre problemas de equilibrio en el  eje real. Aplicaciones

Abstract: En esta charla repasamos algunos conceptos básicos sobre medidas de  equilibrio en presencia de un campo externo, especialmente en el caso en  que el conductor es el eje real. En particular, nos centraremos en lo  que sucede cuando el campo externo es racional, es decir, cuya derivada  es una función racional. En este sentido, presentaremos algunos  resultados recientes sobre el soporte de la medida de equilibrio en  dichas situaciones. Asimismo, comentaremos algunas aplicaciones, bien  conocidas, al estudio asintótico de Polinomios Ortogonales y de  Heine-Stieltjes y a la teoría de Matrices Aleatorias.

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Thursday, January 14 2016, 16:00. Room 2.2.D08

Speaker: Miguel Tierz (Universidade de Lisboa, Portugal)

Title: On Mordell integrals, Stieltjes-Wigert polynomials and some of its physical applications

Abstract: In recent years, a random matrix description of several supersymmetric gauge theories has been developed. This has lead to a situation where a large number of physical observables of such theories admits a very concrete description in terms of integrals. However, while much simpler than the original path-integral description, the integral representations still require exact analytical characterization. Orthogonal polynomials and determinants of Hankel or Toeplitz type are powerful tools for this task. To showcase this, we will solve an explicit example, involving a supersymmetric Chern-Simons theory, by using Mordell integrals and Stieltjes-Wigert polynomials. The mathematical relationship between the two will be addressed as well.

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Thursday, November 5 2015, 16:00. Room 2.2.D08

Speaker: Edmundo Huertas (Universidad de Alcalá)

Title: Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure.

Abstract: This paper deals with monic orthogonal polynomials generated by the so called Geronimus canonical spectral transformation of the Laguerre classical measure, namely one of the three canonical perturbations of measures on the Real line.  We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as n tends to infinity. This is a joint work with Alfredo Deaño and Pablo Román.

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Thursday, October 29 2015, 16:00. Room 2.2.D08

Speaker: Yves Grandati (Université de Lorraine, France)

Title: Shape invariance and equivalent mixed Jacobi-Trudi formulas forexceptional orthogonal polynomials

Abstract: On the basis of shape invariance arguments and specific discrete symmetries of the harmonic and isotonic potential, we obtain Jacobi-Trudi type formulas for the X-Hermite and X-Laguerre polynomials and we describe the equivalence relations among these determinantal representations.

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Thursday, October 22 2015, 16:00. Room 2.2.D08

Title: Non-linear mixed norm spaces for the critical Sobolev embedding.

Abstract: During the 1930s Sobolev  introduced the spaces that now bear his name and, at the same time, he  proved his classical theorem that W10 Lp(In) embeds into Lp*(In),where p*=pn/(n-p). When we let p tend to n from the left, then p* tends to ∞. However, one can have unbounded functions in W10 Ln(In). Consequently,  it is necessary to go outside the Lebesgue scale to  find the optimal conditions satisfied by functions in W10 Ln(In).

An early result in this direction was obtained by Trudinger and was subsequently improved and generalized by Hansson, Brezis and Wainger and Maz'ya who obtained the following Lorentz-type refinement  for the so-called limiting case of Sobolev embedding:
W10 Ln(In) embedds into L∞,n;-1 (In).

Although the above estimate is the best possible as far as rearrangement invariant range spaces are concerned, Bastero, Milman, and Ruiz, and independently Maly and Pick, proved that if the requirement that the target space should be a linear space is abandoned, then a further improvement of this borderline case is still possible. Motivated by these works, our goal is to  reformulated their results in terms of non-linear mixed norm spaces.

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Thursday, October 15 2015, 16:00. Room 2.2.D08

Speaker: Daniel Seco (Universidad de Barcelona)

Title: Ciclicidad contra polinomios ortogonales.

Abstract: En un espacio de funciones holomorfas Y, una función f es cíclica si al multiplicarla por todos los polinomios se obtiene un subespacio X, denso en Y. En un trabajo anterior con varios colaboradores, introdujimos el uso de herramientas de optimización para encontrar explícitamente polinomios que determinan la ciclicidad de una función, llamados aproximantes óptimos. Ahora, encontramos una correspondencia entre estos aproximantes óptimos y familias de polinomios ortogonales en X, y explotamos esto para obtener varias caracterizaciones nuevas de la ciclicidad y propiedades de los ceros de los aproximantes óptimos.

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Thursday, October 8 2015, 16:00. Room 2.2.D08

Speaker: Jorge Arvesú (Universidad Carlos III de Madrid)

Title: nth root asymptotics for multiple Meixner polynomials.

Abstract: The nth root asymptotic behavior of multiple Meixner polynomials is presented. Two main ingredients of the proposed approach for the study of the aforementioned asymptotic behavior are used and discussed; namely, an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution and the limiting behavior of the coefficients of the recurrence relations.

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Thursday, September 24 2015, 16:00. Room 2.2.D08

Speaker: Francesco Calogero (Università di Roma "La Sapienza", Italia)

Title: On the generations of monic polynomials obtained by replacing the coefficients of the polynomials of the next generation with the zeros of a polynomial of the previous generation.

Abstract: Generations of monic polynomials ---all of arbitrary degree N--- are obtained from a seed polynomial of degree N by identifying the coefficients of the polynomials of the next generation with the zeros of a polynomial of the previous generation; and Diophantine properties are reported of the zeros of the polynomials therebyobtained when the seed polynomial is the Hermite polynomial of degree N. This is joint work in progress with Oksana Bihun.

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Last Updated on Wednesday, 14 September 2016 09:36

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