Multivariate Orthogonal Polynomials and Modified Moment Functionals

TL;DR

This paper explores how modifications to the moment functional, such as adding mass points or multiplying by a polynomial, affect multivariate orthogonal polynomials and their properties, with illustrative examples.

Contribution

It provides a detailed analysis of Uvarov and Christoffel modifications on multivariate orthogonal polynomials and examines their impact on polynomial properties.

Findings

Characterization of orthogonal polynomials under modifications

Impact of modifications on polynomial properties

Illustrative examples demonstrating theoretical results

Abstract

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained by adding to the moment functional a finite set of mass points, or by multiplying it times a polynomial of total degree 2, respectively. Orthogonal polynomials associated with modified moment functionals will be studied, as well as the impact of the modification in useful properties of the orthogonal polynomials. Finally, some illustrative examples will be given.