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

Abstract

For every system $\left\{p_n\right(z){\}}_{n=0}^{\mathrm{\infty }}$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n\left(z\right)$, 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 z) are given. We define suitable Sobolev spaces with matrix weights and consider measurable factorizations of weights. 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.

Keywords:

• Sobolev orthogonal polynomials
• generating function
• integral representation
• recurrence relation
• differential equation
• asymptotics

2010 Mathematics Subject Classification:

• 42C05

Acknowledgements

The author thanks the referees for their valuable comments and suggestions. One of the referees proposed to replace the eigenvalue in (2(2) $L\overrightarrow{y}\left(z\right)=zM\overrightarrow{y}\left(z\right),\overrightarrow{y}\left(z\right)=\left(y_0\right(z),y_1(z),\dots {)}^T,$(2) ) by a function of z, and then Problem 3.1 appeared.

Disclosure statement

No potential conflict of interest was reported by the author(s).