On the entropy for indeterminate moment problems

TL;DR

This paper investigates the entropy properties of densities solving indeterminate Hamburger moment problems, showing they all have finite Shannon entropy and are either uniformly bounded or unbounded, with an example illustrating the bounded case.

Contribution

It proves that all densities in the family solving an indeterminate Hamburger moment problem have finite Shannon entropy and are uniformly bounded or unbounded, extending understanding of their analytic properties.

Findings

All densities have finite Shannon entropy.

Densities are either all bounded or all unbounded.

The Al-Salam--Carlitz moment problem exemplifies the bounded case.

Abstract

For an indeterminate Hamburger moment problem we consider an infinite family of analytic densities solving the moment problem and we prove that they all have finite (Shannon) entropy. These densities are either all bounded or all unbounded. The result is illustrated by the Al-Salam--Carlitz moment problem, where all the densities in the family are bounded.