Moment-sequence transforms

TL;DR

This paper characterizes functions that preserve the moment sequences of positive measures on the real line, revealing their structure and implications for Hankel matrices and multivariable transforms.

Contribution

It provides a complete classification of moment-preserving functions, showing they are built from absolutely monotonic components and characterizing their effects on Hankel and totally non-negative matrices.

Findings

Functions preserving moments are composed of absolutely monotonic functions.

Preserving moments of three-point measures implies preservation of all measures.

Transforms preserving totally non-negative Hankel matrices are constant or linear.

Abstract

We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.