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Weekly Seminar
 WS Season 2008/2009

# Seminar on Orthogonality, Approximation Theory and Applications.

1. Thursday. June 25, 2009. 16:00. Room: 2.2.D.08.
• Title: Strong asymptotic on mixed Nikishin-Jacobi polynomials.
• Speaker: Ulises Fidalgo Prieto, U. Carlos III de Madrid, Spain.
• Abstract: Using Fokas', Its' and Kitaev's approach, we are going to present some relationships between Riemann-Hilbert problems and different kinds of orthogonality of polynomials. With the known steepest decent method, those Riemann-Hilbert problems allow strong asymptotic expressions for sequences of orthogonal polynomials to be found (here the orthogonality is understood in a wide sense). Our talk is going to be focused in obtaining strong asymptotic expressions for sequences of mixed type of multiple orthogonal polynomials corresponding to two Nikishin systems. Both of these two Nikishin systems are generated by systems of measures with type Jacobi weights. As a motivation, our exposition will start with an analysis of a one-dimensional Brownian movement corresponding to a single particle, where a strong asymptotic expression for a sequence of Jacobi polynomials plays a fundamental role.
2. Thursday. June 18, 2009. 16:00. Room: 2.2.D.08.
• Title: A numerical solution of the constrained weighted energy problem and its relation to rational Lanczos iterations.
• Speaker: Karl Deckers, Departament of Computer Science, K. U. Leuven (Belgium).
• Abstract: A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem are studied. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of the eigenvalues, on the distribution of the poles, and on the ratio between the size of the matrix and the number of iterations. The algorithm presented gives the possibility to find the boundary of this region in an effective way. We give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational Lanczos algorithm.
3. Thursday. June 4, 2009. 16:00. Room: 2.2.D.08.
• Title: Relative Asymptotics For Orthogonal Matrix Polynomials: Degenerate Case.
• Speaker: Hossain. O. Yakhlef, U. Carlos III de Madrid, Spain.
• Abstract: We study the asymptotic behaviour of the ratio of two sequences of orthonormal matrix polynomials Pn(x;α) and Pn(x;β) (the matrix of measures dβ = dα+Mδ, with δ is the dirac measure) when the matrix parameters {An(α), Bn(α)} in the three-term recurrence relation are convergent (we assume A= lim An to be singular) or are unbounded (we consider the degenerate case).
4. Thursday. may 28, 2009. 16:00. Room: 2.2.D.08.
• Title: Mixed type of multiple orthogonal polynomilas for two Nikishin systems. (Prelectura de tesis doctoral).
• Speaker: Abey López García, Depatament of Mathematics, Vanderbilt University, Nashville, Tn., U.S.A.
• Abstract: We study logarithmic and ratio asymptotic of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated from another Nikishin system. This condition combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotic of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotic is described by means of a conformal representation of an appropiated Riemann surface of genus zero onto the extended complex plane.
5. Thursday. may 21, 2009. 16:00h. Room: 2.2.D.08.
• El operador de multiplicación, localización de ceros y asintótica de polinomios extremales para normas de Sobolev no diagonales.
• Speaker: Eva Touris Lojo, U. Carlos III de Madrid.
• Abstract: En un trabajo previo de Lagomasino, Pérez Izquierdo y Pijeira se estudia la localización de ceros y el comportamiento asintótico de polinomios extremales para normas de Sobolev no diagonales (es decir, en la norma aparecen productos de la función y de sus derivadas de diferentes órdenes). Los polinomios ortogonales de Sobolev con respecto a dicha norma son un caso particular de dichos polinomios extremales (cuando p=2). Creemos que el enfoque de Lagomasino, Pérez Izquierdo y Pijeira es el más apropiado para el estudio del caso general (admitiendo derivadas hasta orden N). Sin embargo, si nos centramos en el caso N=1 (sólo aparecen primeras derivadas), que es el caso más usual, conseguimos una doble mejora de sus resultados: por un lado, debilitamos las hipótesis técnicas y, por otro, nuestras hipótesis involucran directamente los coeficientes de la matriz, en lugar de sus autovalores.
6. Thursday. may 14, 2009. 16:00h. Room: 2.2.D.08.
• Runge-Kutta convolution quadrature methods and transparent boundary conditions (Colloquium).
• Speaker: Cesar Palencia de Lara.
• Abstract: Recent results concerning Runge-Kutta based convolution quadrature methods for abstract, well posed, linear, and homogeneous Volterra equations show a general representation of the numerical solution in terms of the continuous one. The interest of such a representation goes beyond the error and
stability analysis, since it explains that the numerical solution, under suitable methods, inherits some important qualitative properties, such as positivity or contractivity, of the exact solution. In this talk we will summarize this theory and use it for the construction of fully discrete, transparent boundary conditions, for the heat and Schroedinger equation, with good qualitative properties.
7. Thursday. may 7, 2009. 12:00. Room: 2.2.D.08.
• Title: El problema clásico de momentos, la aproximación racional y los polinomios ortogonales. (Colloquium)
• Speaker: Manuel Bello Hernández, Dpto. de Matemática y Computación, U. de la Rioja, (Spain).
• Abstract: El problema clásico de momentos es muy importante en los aportes al análisis clásico en el período que va desde los trabajos de Stieltjes hasta los de Krein. La noción de medida (integral de Stieltjes), los aproximantes de Padé, los polinomios ortogonales, los métodos de cuadratura, la extensión de funcionales lineales positivos (teorema de Riesz) y los problemas de valores frontera de funciones analíticas tienen sus raíces en el estudio del problema clásico de momentos. En la charla veremos como la aproximación racional unilateral está directamente relacionada con la solución del problema clásico de momentos. La conexión anterior se utilizará en el estudio de la asintótica relativa de polinomios ortogonales y de la convergencia de aproximantes de Padé de funciones que son la suma de una fracción racional y la transformada de Cauchy de una medida positiva con soporte en la recta real. En esta dirección se presentarán resultados que extienden teoremas obtenidos previamente por Guillermo López.
8. Thursday. april 30, 2009. 16:00. Room: 2.2.D.08.
• Title: Sobolev type orthogonality: a hydrodynamical interpretation.
• Speaker: Hector Pijeira Cabrera, Dpto. de Matemática, U. Carlos III de Madrid, (Spain).
• Abstract: We study the algebraic (Zeros location), differential and asymptotic properties of orthogonal polynomials with respect to discrete--continuous Sobolev type inner product with respect to Gegenbauer measures, these polynomials are also primitives of Gegenbauer polynomials. A hydrodynamic model for source points location of a flow of an incompressible fluid with preassigned stagnation points on [-1,1] is solved.
9. Thursday. April 23, 2009. 16:00. Room: 2.2.D.08.
• Title: Sistemas integrables y polinomios ortogonales matriciales.
• Speaker: Carlos Álvarez Fernández, Departamento de Física Teórica II, Facultad de Ciencias Físicas, U.C.M. (Spain).
• Abstract: Se explorarán las conexiones entre las reducciones Toeplitz-Hankel de la jerarquía de Toda multicomponente y las familias de polinomios biortogonales y ortogonales matriciales en la recta y en el círculo. Igualmente se tratarán casos de ortogonalidad generalizada y sus problemas asociados (problemas de Riemann-Hilbert y relaciones de recurrencia).
10. Thursday. April 16, 2009. 16:00. Room: 2.2.D.08.
• Title: On linear forms containing Euler constant (colloquium,).
• Speaker: Alexader Aptekarev, Keldysh Institute of Applied Mathematics, Moscow (Russia).
• Abstract: We discuss arithmetical properties of mathematical constants related with infinite summs of the inverse powers, i.e. the Euler constant, the values of the Riemann Zeta-function at the odd points. In 1978 Aperi has proved that Zeta(3) is irrational number. Since that time there was no much progress achieved in this field. In 2000 Rivoal proved that among the values of the Riemann Zeta-function at the odd points there are infinitely many irrational numbers and in 2001 Zudilin has precised this result proving that one number from Zeta(5), Zeta(7), Zeta(9) and Zeta(11) is irrational. We tell about similar result involving Euler constant.
11. Thursday. april 2, 2009. 16:00. Room: 2.2.D.08.
• Title: Zeros of Heine-Stieltjes polynomials and critical measures (Colloquium).
• Speaker: Andrei Martínez Finkelshtein, Universidad de Almería (Spain).
• Abstract: Heine-Stieltjes polynomials are solutions of certain second order linear ODEs with polynomial coefficients. They generalize the families of classical orthogonal polynomials and arise in several areas of mathematics and phisics. Their zeros are saddle points of certain discrete energy functionals, and the description of their asymptotics requires an extension of the standard notion of equilibrium. We analyze some related potential theoretic concepts, as well as discuss briefly their connection with trajectories of quadratic differentials and extremal problems on the plane. This is a partial report of a joint work with E. Rakhmanov.
12. Thursday. march 26, 2009. 16:00. Room: 2.2.D.08.
• Title: The numerical inversion of the Laplace transform with applications to evolution problems.
• Abstract: Laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting there a potential behavior are considered (sectorial Laplace transforms). A spectral order, parallelizable method for their numerical inversion is proposed. The method takes into account the available information about the errors arising in the evaluations. The application of the Laplace transform and the proposed inversion algorithm to integrate in time evolution equations is considered
13. Thursday. march 12, 2009. 16:00. Room: 2.2.D.08.
• Abstract: Se verán algunos resultados sobre las desigualdades de Hardy-Sobolev "optimales" y constantes optimales (asociadas a dichas desigualdades) bajo distintos marcos funcionales. Finalmente, se presentarán algunas aplicaciones en el marco de las ecuaciones en derivadas parciales elípticas no lineales con distintos tipos de condiciones de contorno. El contenido de la Charla forma parte de los siguientes astículos:
• E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions. J. Funct. Anal., 2003.
• B. Abdellauoi, E. Colorado, I. Peral, Some remarks on elliptic equations with singular potentials and mixed boundary conditions. Adv. Nonlinear Stud., 2004.
• B. Abdellauoi, E. Colorado, I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities. Cal. Var. Partial Differential Equations, 2005.
• B. Abdellauoi, E. Colorado, I. Peral, Effect of the boundary conditions in the behavior of the optimal constant of some Caffarelli-Kohn-Nirenberg inequalities. Application to some doubly critical nonlinear elliptic problems. Adv. Differential Equations, 2006.
14. Thursday. march 5, 2009. 16:00. Room: 2.2.D.08.
• Title: Metric graphs and boundary triples (Colloquium).
• Speaker: Olaf Post, Humbodt Universitaet Berlin (Germany).
• Abstract: We use the notion of boundary triples in order to analyse second order differential operators on metric graphs. Boundary triples are an abstract formulation of Green's formula for the Laplacian on a domain with boundary. In particular, for equilateral graphs we obtain a simple relation of a metric graph differential operator with a discrete analogue acting as difference operator on the vertices. The latter difference operator generalises the usual combinatorial discrete Laplacian.
15. Thursday. february 26, 2009. 16:00. Room: 2.2.D.08.
• Title: Nikishin systems are perfect.
• Abstract: A long standing open problem in the theory of multiple orthogonal polynomials is whether or not Nikishin systems are perfect. That is, whether the corresponding multiple orthogonal polynomials are of maximum possible degree (or uniquely determined by the orthogonality relations they satisfy) no matter what multi-index you take. This question is of great importance in applications of Hermite-Padé approximation to number theory, convergence theory of simultaneous approximation, and in extending the asymptotic theory of standard orthogonal polynomials to the case of multiple orthogonality. Hermite himself proved that systems of exponentials are perfect and this plays a fundamental role in his proof of the trascendence of number $e$. We give a positive answer to the conjecture. The proof relies on a property which is of independent interest and in a sense extends the fundamental theorem of algebra to generalized polynomials which can be expressed as polynomial combination of Markov functions generated by a Nikishin system. We also prove that the zeros of these multiple orthogonal polynomials are simple and those of "consecutive" multi-indices interlace.
16. Thursday. february 19, 2009. 16:00.Room: 2.2.D.08.
• Title: Geodésicas que escapan a infinito: azar y necesidad.
17. Thursday. february 12, 2009. 16:00. Room: 2.2.D.08.
• Title: Resultados sobre aproximación e interpolación simultánea.
• Speaker: José G. LLavona, Departamento de Análisis Matemático, UCM, Madrid (Spain).
• Abstract: En los comienzos de los años 1970 aparece el interés de extender, al caso infinito dimensional, los clásicos teoremas de aproximación para funciones definidas en Rn.Vamos a tratar este tema en relación al Teorema de Weierstrass, y como aplicación de los resultados conseguidos, presentaremos un teorema de aproximación e interpolación simultánea de funciones diferenciables por polinomios.
Referencias
J. G. Llavona, Approximation of differentiable functions, Advances in Math., Supplementary Studies 4 (1979) 197-221.
J. G. Llavona, Approximation of continuously differentiable functions, North-Holland Mathematics Studies, 130. Notas de Matemática (112). Amsterdam, 1
18. Thursday. february 5, 2009. 16:00. Room: 2.2.D.08.
• Title: Reconfiguration of cube-style modular robots.
• Speaker: Vera Sacristán Dept. Matemàtica Aplicada II, U. Politècnica de Catalunya.
• Abstract: I will present some recent results and ongoing research on self-reconfiguration of modular robots composed of cubical units (atoms) arranged in a lattice configuration and capable of performing four basic actions: expand and contract, as well as attach and detach from neighbors. This work includes centralized and distributed solutions, as well as massive parallelization of the robot moves. The proposed algorithms are in-place, guarantee the connection of the reconfiguration space and improve the previously known algorithms. This is joint work with G. Aloupis, S. Collette, M. Damian, E. Demaine, R. Flatland, S. Langerman, J. O'Rourke, S. Ramaswami and S. Wuhrer. And partially also with D. El-Khechen and V. Pinciu, as well as with O. Aichholzer, T. Hackl and B. Vogtenhuber.
19. Thursday. January 29, 2009. 16:00. Room: 2.2.D.08.
• Title: Critical points and level Sets in the exterior boundary problems Conferenciante: Daniel Peralta Salas, Universidad Carlos III de Madrid.
• Abstract: We study some geometrical properties of the critical set of the solutions to an exterior boundary problem in Rn\(Ω U δΩ), where Ω is a bounded domain with C2 connected boundary. We prove that this set can be nonempty (in fact, of codimension 3) even when Ω contractible, thereby settling a question posed by Kawohl. We also obtain new sufficient geometric criteria for the absence of critical points in this problem and analyze the properties of the critical set for generic domains. The proofs rely on a combination of classical pothential theory, transversality techniques and the geometry of real analytic sets..
20. Thursday. January 22, 2009. 16:00. Room: 2.2.D.08.
• Title: Calculating Rational Best Approximation. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
• Speaker: Herbert Stahl, THF-Berlin, Germany.
• Abstract: We shall present a new algorithm for the calculation of rational best approximants to real functions on a real interval. The development of the algorithm has been motivated by a need to calculate rational best approximants on (-∞,0] to functions that are similar to the exponential function, as for instance, rational perturbations of this function. The talk will be opened with several examples of this kind. After that, the basic structure of the new algorithm will be explained (it shares several aspects with the Remez algorithm), and we then have a closer look on some of its key elements.
21. Thursday. January 15, 2009. 16:00. Room: 2.2.D.08.
• Title: Soluciones aproximadas del sistema de EDO que define las geodésicas.
22. Thursday. December 11, 2008. 16:00. Room: 2.2.D.08.
• Title: El problema de la integración en forma cerrada de Ecuaciones Diferenciales. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
• Abstract: En esta charla se discutirá el problema de la integración en forma cerrada de ecuaciones diferenciales ordinarias. Nuestra tesis, que ilustraremos mediante ejemplos concretos, es que -a diferencia de lo que se suele creer- no es la teoría de Lie de simetrías la que es más útil para la integración mediante métodos analíticos de una ecuación diferencial, sino la teoría de Galois de ecuaciones diferenciales.
23. Thursday. December 4, 2008. 16:00. Room: 2.2.D.08.
• Title: El problema Isoperimétrico en los Espacios Producto. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
• Abstract: El problema que da título a esta charla es clásico: encerrar un volumen dado utilizando la menor cantidad posible de área. Ya para el espacio Euclídeo conduce a problemas de Análisis no triviales. Pero además tiene perfecto sentido cuando el espacio ambiente es una variedad Riemanniana, y entonces conduce a nuevos problemas. El problema de estabilidad busca una superficie que al menos sea un mínimo local del área entre todas las que encierran la misma "cantidad de aire". Esto es necesario para que podamos realizar tal superficie como una pompa de jabón. Los dos problemas están muy relacionados, pues una superficie isoperimétrica es un mínimo global. Mostraremos una solución completa a ambos problemas, para volumen grande, cuando el espacio ambiente es el producto de cualquier variedad compacta por un espacio Euclídeo.
24. Thursday. November 27, 2008. 16:00. Room: 2.2.D.08.
• Title: El uso de bases y "frames" en Teoría de Muestreo.
• Abstract: En el seminario repasaremos, brevemente, los conceptos de base ortonormal, base de Riesz y frame en un espacio de Hilbert separable H, poniendo de manifiesto las propiedades que se usarán, posteriormente, en su aplicaci on a la Teoría de Muestreo. En particular, aquellas que nos permiten la recuperaci "estable" de cada x de H a partir de la sucesión {n>} (con n en N) de l2(n). Se justicará también la necesidad del uso de bases de Riesz y de frames, ya que las bases ortonormales se muestran insuficientes en el estudio de muchas cuestiones matemáticas con importancia no sólo teórica sino también aplicada.
La teoría general anteriormente expuesta se utilizará para obtener teoremas de muestreo en los espacios clásicos de Paley-Wiener, así como en espacios invariantes por traslación en L2(R), i.e., subespacios cerrados de L2(R) generados por las traslaciones en los enteros de una cierta función f de L2(R).
25. Thursday. November 20, 2008. 16:00. Room: 2.2.D.08.
• Title: Orthogonal polynomials in several variables and spectral theory of partial differential operators.
• Abstract: Following an historical approach based on the extension of the Routh-Bochner characterization of classical orthogonal polynomials in one variable (Hermite, Laguerre, Jacobi,and Bessel) , we present a constructive approach of some families of two variable orthogonal polynomials which are eigenfunctions of second order partial differential operators with polynomial coefficients. Then, using standard techniques for the symmetrization of partial differential operators, we can deduce the weight function as well as the corresponding domain of orthogonality. In the more general framework of the orthogonality associated with moment functionals on the linear space of polynomials in two variables with real coefficients, classical orthogonal polynomials are defined in terms of a matrix analogue of the Pearson differential equation that such a functional satisfies. They can also be characterized as the polynomial solutions of a matrix second order partial differential equation.
26. Thursday. November 13, 2008. 16:00. Room: 2.2.D.08.
• Title: Orthogonal polynomials and stochastic processes. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
• Speaker: F. Alberto Grunbaum, Math Dept UC Berkeley, U.S.A.
• Abstract: I will discuss a few examples (as well as a general scheme) where orthogonal polynomials and their related spectral analysis play a useful role in probability theory. My main emphasis is in looking for ways to extend this analysis to new situations.
27. Thursday. November 6, 2008. 16:00. Room: 2.2.D.08.
• Title: Spectral aproximation in discrete and quantum graphs.
• Abstract: In this talk we discuss the notion of Laplacians for discrete and quantum (or metric) graphs and mention some spectral relations between these operators. Using this result and a bracketing procedure for eigenvalues we consider the gap structure for infinite graphs. We conclude mentioning some examples where both, the discrete and the quantum graph, have spectral gaps.
28. Thursday. October 30, 2008. 15:30. Room: 2.2.D.08.
• Title: On the multilinear Hausdorff problem of moments.
• Abstract: The multilinear Hausdorff problem of moments will be discussed. Moreprecisely, given a family of moments $\mu_{k_1, \ldots, k_n}$, $k_1, \ldots, k_n = 1, 2, \ldots$, under which conditions they define an integral $n$-linear functional $L$ on the space $C[0,1]$. We will characterize families of moments determining integral $n$-linear functionals by means of a new weak uniform boundedness condition and we will compare it with the classical solution to the (linear) Hausdorff problem. Some problems related with the theory of biorthogonal polynomials will be discussed in the bilinear case.
29. Thursday. October 23, 2008. 16:00. Room: 2.2.D.08. COLLOQUIUM DEPARTAMENTO DE MATEMÁTICAS.
• Title: Convergent Interpolation to Cauchy Transforms.
• Speaker: Maxim Yattselev, INRIA, Sophia Antipolis, France.
• Abstract: We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini–smooth non–vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non–Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicitgranted the parametrization of the arc.. (joint work with L. Baratchart).
30. Thursday. October 16, 2008. 16:00. Room: 2.2.D.08.
• Title: Aproximantes y polinomios multi-ortogonales mixtos.
• Abstract: Se presentara la conexion entre la aproximacion Hermite-Pade mixta (en la cual se combina tipo I y tipo II) en un marco general y los polinomios multiortogonales mixtos. Se particularizara en el caso en que la aproximacion mixta provenga de sistemas de Nikishin y se presentara algunos resultados recientes sobre la asintotica logaritmica y del cociente para polinomios multi-ortogonales mixtos de Nikishin.
31. Thursday. October 9, 2008. 16:00. Room: 2.2.D.08.
• Title: Strong asymptotic for eigenpolynomials of a class of differential operators.
• Abstract: We give the strong asymptotic behavior of a family of polynomials which are
eigenfunctions of a class of differential operators, using the WKB method.
(joint work with H. Pijeira).
32. Thursday. October 2, 2008. 16:00. Room: 2.2.D.08.
• Title: Neoclassical orthogonal polynomials.
• Abstract: The classical orthogonal polynomial systems (OPS) of Jacobi, Laguerre and Hermite are traditionally characterized as (the only) polynomial solutions of a Sturm-Liouville problem, after the classical results of Bochner and Lesky. We show that other complete Sturm-Liouville OPS exist if the sequence of polynomial eigenfunctions is not required to start with a constant. In this talk we will introduce these polynomial families and study some of their basic properties, which closely resemble those of the clasical families.
This is joint work with Niky Kamran (Mc Gill) and Robert Milson (Dalhousie).
33. Thursday. September 25, 2008. 16:00. Room: 2.2.D.08.
• Title: Approximation problems in Systems Theory.
• Speaker: Diederich Hinrichsen, University of Bremen, Germany.
• Abstract: Se describen algunos problemas de aproximación que surgen en la Teoría de Sistemas y presento métodos para su solución (en particular la teoría de Adamyan-Arov-Krein). Explicaré algunos conceptos de la teoría de sistemas que sirven de soporte para estos métodos y discuto su relación con resultados de la teoría de operadores y la teoría de aproximación.

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