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On the entropy for indeterminate moment problems

Christian Berg

TL;DR

The paper investigates Shannon entropy in indeterminate Hamburger moment problems via the Nevanlinna parametrization, proving that the analytic densities $f_{t+i\gamma}$ have finite entropy and that the entropy map is continuous in the parameter. It demonstrates a dichotomy: the entire family of densities is either bounded or unbounded, with boundedness implying $H[f]>-\infty$, and it provides concrete results in the Al-Salam--Carlitz case with a lower bound for the key quantity that ensures finiteness of entropy. The findings extend the understanding of entropy beyond maximum-entropy measures and illustrate the entropy structure with explicit densities and their Krein/Friedrichs connections.

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.

On the entropy for indeterminate moment problems

TL;DR

The paper investigates Shannon entropy in indeterminate Hamburger moment problems via the Nevanlinna parametrization, proving that the analytic densities have finite entropy and that the entropy map is continuous in the parameter. It demonstrates a dichotomy: the entire family of densities is either bounded or unbounded, with boundedness implying , and it provides concrete results in the Al-Salam--Carlitz case with a lower bound for the key quantity that ensures finiteness of entropy. The findings extend the understanding of entropy beyond maximum-entropy measures and illustrate the entropy structure with explicit densities and their Krein/Friedrichs connections.

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.

Paper Structure

This paper contains 2 sections, 3 theorems, 48 equations.

Table of Contents

  1. Introduction and main results
  2. The Al-Salam--Carlitz moment problem

Key Result

Theorem 1.1

For an arbitrary indeterminate Hamburger moment problem the densities $f_{t+i\gamma}$ have entropy $H[f_{t+i\gamma}]\in\mathbb{R}$ and the function $t+i\gamma\mapsto H[f_{t+i\gamma}]$ is continuous from $\mathbb{H}$ to $\mathbb{R}$.

Theorems & Definitions (10)

  • Theorem 1.1
  • proof
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3