The Problem of Moments: A Bunch of Classical Results With Some Novelties

TL;DR

This paper reviews classical results on moment (in)determinacy of measures, emphasizing geometric interpretations, eigenvalue bounds, and providing new insights, proofs, and numerical illustrations in the context of Hamburger and Stieltjes moment problems.

Contribution

It introduces novel geometric perspectives, eigenvalue bounds, and alternative proofs that enhance understanding of moment (in)determinacy, along with new numerical examples.

Findings

New lower bounds for smallest eigenvalues of Hankel matrices

Geometric interpretation of indeterminacy conditions

Confirmation of classical results with new arguments

Abstract

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well-known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger's results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices; these bounds are used for concluding indeterminacy; (c) provide new arguments to confirm classical results; (d) give new numerical illustrations involving commonly used probability distributions.