On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

TL;DR

This paper constructs Sobolev orthogonal polynomials related to classical orthogonal polynomials, providing explicit integral representations and exploring their eigenvalue properties and applications in function systems.

Contribution

It introduces explicit integral representations for Sobolev orthogonal polynomials based on classical polynomials and identifies new families with eigenvalue properties.

Findings

Explicit integral representations involving classical orthogonal polynomials.

Two new families of Sobolev orthogonal polynomials with eigenvalue properties.

Connections between Sobolev orthogonal polynomials and orthogonal systems in direct sum spaces.

Abstract

For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in $n$) and generalized eigenvalues of a differential operator (in $n$) are given. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar $L^2_\mu$ spaces are discussed.