Orthogonal matrix polynomials and differential, difference and q-difference matrix operators

Texto completo descargado desde TeseoIn this thesis we present a series of result that are framed in the theory of Matrix Orthogonal Polynomials, a branch of the very celebrated subject of Orthogonal Polynomials. In particular we study families of matrix valued orthogonal polynomials satisfying differential, difference or q-difference equations. The search for examples of matrix polynomials which are also eigenfunctions of certain matrix operators is a rather dificult issue. This dificulty is due to several reasons. The most important of these reasons is the increase of the difficulties in the computations related with the non-commutativity of the matrix product, and the existence of singular matrices. However, having a wide set of examples of matrix orthogonal polynomials has shown to be decisive in the study and discovery of new phenomena happening in the matrix orthogonality. This has been the case with the examples of matrix orthogonal polynomials satisfying differential equations. An example of the increasing knowledge about these families of orthogonal polynomials is shown in the last chapter of this memory. There, lot of tools developed in the last decades are used to build and study in deep an interesting family of matrix orthogonal polynomials satisfying second order differential equations. Moreover, these families are shown to satisfy first order differential equations as well. In the case of matrix orthogonal polynomials satisfying difference equations, very little was known. Apart from some examples in size 2x2 (and some others reducible to the scalar case) there were no examples of such matrix orthogonal polynomials. With this thesis this lack of examples starts to be solved. Moreover, we introduce a method to construct examples of matrix orthogonal polynomials satisfying second order difference equations, and by making use of it we give a variety of examples. Having such a method is of the main importance, because we skip the complexity in the computations that made the search of examples so difficult, and now dealing with matrix orthogonal polynomials and matrix difference equations becomes much easier to handle. The method profits of the factorization of a weight matrix and the symmetry equations for a discrete weight matrix and a difference operator. These symmetry equations are the starting point to develop the method. By making use of the examples obtained by this method, we explore new features and properties satisfied by this objects. For the case of matrix orthogonal polynomials satisfying q-difference equations, even less was known. In this thesis we establish the symmetry equations for the q-difference case, and we adapt the method developed for the difference case to obtain the first non-trivial examples of matrix orthogonal polynomials satisfying second order q-difference operator. That emphasizes the power of the method to build examples for the difference case. With our method we construct an example of matrix orthogonal polynomials satisfying q-difference equations, but this method can easily be used to get a wider class of examples and to explore their properties. With the content of this thesis we get a more complete view of the theory of matrix orthog- onal polynomials, and many questions can now be tackled, such as those concerning limiting relations among matrix orthogonal polynomials satisfying second order difference equations (or q-difference) and matrix orthogonal polynomials satisfying second order differential equa- tions

SU(2) WZW Theory at Higher Genera

We compute, by free field techniques, the scalar product of the SU(2) Chern-Simons states on genus > 1 surfaces. The result is a finite-dimensional integral over positions of ``screening charges'' and one complex modular parameter. It uses an effective description of the CS states closely related to the one worked out by Bertram. The scalar product formula allows to express the higher genus partition functions of the WZW conformal field theory by finite-dimensional integrals. It should provide the hermitian metric preserved by the Knizhnik-Zamolodchikov-Bernard connection describing the variations of the CS states under the change of the complex structure of the surface.Comment: 44 pages, IHES/P/94/10, Latex fil

Polinomios ortogonales confluentes matriciales

El presente trabajo se enmarca en la teoría de polinomios ortogonales matriciales que satisfacen una ecuación diferencial de tipo hipergeométrico. Debido a la no conmutatividad de matrices y a la existencia de matrices singulares, en esta teoría surgen interesantes fenómenos ausentes en el caso de los polinomios ortogonales clásicos. Mientras que en este sólo existen tres familias distintas que son ortogonales respecto a un peso positivo, en el caso matricial la cantidad de familias es infinita. Además, esta teoría se caracteriza por la existencia de varias familias distintas de polinomios ortogonales matriciales que son autofunciones de un mismo operador diferencial de segundo orden, o de varios operadores diferenciales de segundo orden que tienen a una misma familia de polinomios ortogonales matriciales como autofunción. El objetivo de este trabajo consistió en encontrar familias de polinomios ortogonales mónicos matriciales {Pn}n∈N0 de tamaño 2×2, que son autofunciones del operador hipergeométrico confluente matricial.Fil: Torres, Analía Victoria . Universidad Nacional de Cuyo. Facultad de Ciencias Exactas y Naturales

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems