Krein condition and the Hilbert transform

TL;DR

This paper leverages the Hilbert transform to construct families of densities sharing the same moments as a given density, providing new insights into M-indeterminacy and Stieltjes classes.

Contribution

It introduces a novel method using the Hilbert transform to explicitly construct densities with identical moments, expanding the understanding of M-indeterminacy.

Findings

Constructs families of densities with same moments using Hilbert transform

Provides explicit Stieltjes classes centered at a given density

Addresses densities supported on entire real line or positive half-line

Abstract

Krein condition have been used as a qualitative result to show the M-indeterminacy of some kind of densities. In this work we use results from the theory of the Hilbert transform to construct families of densities having all the same finite moment sequence as a density $f$ with finite logarithmic integral. Actually, our approach explicitly gives Stieltjes classes with center at $f$ and perturbations involving the Hilbert transform of $\ln f.$ We consider densities supported on the whole real line or the positive half line.