From asymptotics to spectral measures: determinate versus indeterminate moment problems

TL;DR

This paper explores the differences between determinate and indeterminate moment problems in orthogonal polynomials, highlighting how spectral measures are constructed and analyzed using asymptotics and Nevanlinna theory, with examples involving elliptic functions.

Contribution

It provides a comparative analysis of spectral measure construction methods for determinate and indeterminate moment problems, including new examples related to elliptic functions.

Findings

Spectral measures for determinate problems are governed by polynomial asymptotics.

Indeterminate problems require Nevanlinna theory for spectral measure construction.

Weaker forms of Markov theorem can still produce some spectral measures.

Abstract

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna theory. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.