Hessenberg-Sobolev Matrices and Favard type theorems

TL;DR

This paper extends Favard's theorem to Sobolev orthogonal polynomials by characterizing Hessenberg matrices and their moment matrices, establishing a structural link between matrix operators and Sobolev inner products.

Contribution

It provides a novel characterization of Hessenberg matrices associated with Sobolev orthogonality, generalizing classical Favard theorems.

Findings

Characterization of Hessenberg matrices related to Sobolev orthogonality

Extension of Favard's theorem to Sobolev inner products

Structural description of moment matrices in this context

Abstract

We study the relation between certain non-degenerate lower Hessenberg infinite matrices $\mathcal{G}$ and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known Favard theorem for Sobolev orthogonality. We characterize the structure of the matrix $\mathcal{G}$ and the associated matrix of formal moments $\mathcal{M}_{\mathcal{G}}$ in terms of certain matrix operators.