On linearly related orthogonal polynomials in several variables

Abstract

Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$ are constant matrices of proper size and $\mathbb{Q}_0 = \mathbb{P}_0$. The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.Comment: 28 pages. To appear in Numerical Algorithm

Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies

“…Also in this general multidimensional framework we have studied in [15] multivariate Laurent polynomials orthogonal with respect to a measure supported in the unit torus, finding in this case the corresponding Christoffel formula. In [7] linear relations between two families of multivariate orthogonal polynomials were studied. Despite that [7] does not deal with Geronimus formulae, it deals with linear connections among two families of orthogonal polynomials, a first step towards a connection formulae for the multivariate Geronimus transformation.…”

“…In [7] linear relations between two families of multivariate orthogonal polynomials were studied. Despite that [7] does not deal with Geronimus formulae, it deals with linear connections among two families of orthogonal polynomials, a first step towards a connection formulae for the multivariate Geronimus transformation.…”

Multivariate Orthogonal Polynomials and Modified Moment Functionals

“…We will work with the particular case when the polynomial λ(x) has total degree 2. We must remark that in several variables there exist polynomials of second degree that they can not be factorized as a product of two polynomials of degree 1, and then this case is not a trivial extension of the case considered in [2]. Using a block matrix formalism for the three term relations, this case have been also considered in [4] and [5] for arbitrary degree polynomials.…”

“…In Section 4 we study the Christoffel modification by means of a second degree polynomial. This is not a trivial extension of the case when the degree of the polynomial is 1 studied in [2], since in several variables not every polynomial of degree 2 factorizes as a product of polynomials of degree 1. Again, we relate both families of orthogonal polynomials and also we deduce the orthogonality by using Favard's theorem in several variables.…”

“…To our best knowledge, one of the first studies about Christoffel modifications in several variables, multiplying a moment functional times a polynomial of degree 1, was done in [2], where the modification naturally appears in the study of linear relations between two families of multivariate polynomials. Necessary and sufficient conditions about the existence of orthogonality properties for one of the families was given in terms of the three term relations, by using Favard's theorem in several variables [12].…”