Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations

TL;DR

This paper explores the relationship between higher-order recurrence relations of Sobolev-type orthogonal polynomials and matrix factorizations, linking them to Jacobi matrices via Christoffel transformations.

Contribution

It establishes a connection between (2N+1)-banded matrices for Sobolev polynomials and Jacobi matrices from classical orthogonal polynomial theory.

Findings

Higher-order recurrence relations are represented by (2N+1)-banded matrices.

A link is shown between these matrices and Jacobi matrices from Christoffel transformations.

The results deepen understanding of the structure of Sobolev-type orthogonal polynomials.

Abstract

It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N+1)-banded symmetric semi-infinite matrix. In this paper we state the connection between these (2N+1)-banded matrices and the Jacobi matrices associated with the three-term recurrence relation satisfied by the standard sequence of orthonormal polynomials with respect to the 2-iterated Christoffel transformation of the measure.