Determinacy for measures

TL;DR

This paper investigates the moment problem for measures on the real line using entire functions, characterizes measures by Fourier transform restrictions, and extends results on oscillation frequency of spectral gap measures.

Contribution

It introduces a generalized moment problem framework and extends existing theorems on spectral measures with gaps.

Findings

Characterization of measures via Fourier transform restrictions

Extension of Eremenko and Novikov's theorem on oscillations

New criteria for measure determinacy using entire functions

Abstract

We study the general moment problem for measures on the real line, with polynomials replaced by more general spaces of entire functions. As a particular case, we describe measures that are uniquely determined by a restriction of their Fourier transform to a finite interval. We apply our results to prove an extension of a theorem by Eremenko and Novikov on the frequency of oscillations of measures with a spectral gap (high-pass signals) near infinity.