The Sobolev moment problem and Jordan dilations

TL;DR

This paper explores the Sobolev moment problem and Jordan dilations, highlighting recent developments in Sobolev orthogonal polynomials and advocating for further research in their moment counterparts.

Contribution

It revisits the Sobolev moment problem, connecting it with Jordan dilations, and emphasizes the need for deeper exploration of Sobolev orthogonal polynomials' moment theory.

Findings

Recent progress in Sobolev orthogonal polynomials

Identification of gaps in the moment problem for Sobolev spaces

Proposal for future research directions

Abstract

Moment problems and orthogonal polynomials, both meant in a single real variable, belong to the oldest problems in Classical Analysis. They have been developing for over a century in two parallel, mostly independent streams. During the last 20 years a rapid advancement of polynomials orthogonal in Sobolev space has been noticed; their moment counterpart seems to be not paid enough attention as it deserves. In this paper we intend to resume the theme and open the door for further, deeper study of the problem.