November 16, 2007
We give a survey of the analytic theory of matrix orthogonal polynomials.
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June 7, 2011
We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators ...
May 31, 2023
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain information about the matrix orthogonal polynomials and functions of second kind associated with a weight matrix. We deduce properties for the recurrence relation coefficients from differential properties of the weight matrix. We take the matrix polynomials of Hermite, Laguerre and Jacobi type as a case study.
December 18, 2005
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
October 5, 1993
It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. In this paper we extend this result and show that every system of polynomials satisfying some $(2N+1)$-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients...
March 12, 2013
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
October 3, 2006
Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for classical orthogonal polynomials in two variables.
April 26, 2011
We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg\H{o}'s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.
March 15, 2006
In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with an herm...
March 13, 2006
In this paper we describe various applications of the Riemann-Hilbert method to the theory of orthogonal polynomials on the line and on the circle.
December 29, 2004
In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.