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

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 $L_{\mu }^2$ spaces are discussed.

arXiv comment28 pages