Krall--type Orthogonal Polynomials in several variables

TL;DR

This paper derives explicit formulas for multivariable orthogonal polynomials and their reproducing kernels, enabling new inequalities for classical orthogonal polynomials, based on modifications of positive definite moment functionals.

Contribution

It introduces explicit formulas for multivariable orthogonal polynomials and their kernels derived from modified moment functionals, advancing the understanding of Krall-type polynomials.

Findings

Explicit formulas for multivariable orthogonal polynomials

Explicit reproducing kernel formulas

New inequalities for classical orthogonal polynomials

Abstract

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.