Local Versions of the Wiener-Levi Theorem

TL;DR

This paper extends the Wiener-Levi theorem to local settings, demonstrating that for certain measures, the Fourier transform can be composed with a real-analytic function to produce another measure with a prescribed Fourier transform.

Contribution

It introduces local versions of the Wiener-Levi theorem, allowing composition of Fourier transforms with real-analytic functions in a localized context.

Findings

Existence of measures with prescribed Fourier transforms via composition with real-analytic functions.

Extension of classical Wiener-Levi theorem to local and measure-theoretic settings.

Applicable to measures without singular components on the Euclidean space.

Abstract

Let h be a real-analytic function in the neighborhood of some compact set K on the plane. We show that for any complex measure on the Euclidean space of a finite total variation without singular components with the Fourier--Stieltjes transform f(y) there exists another measure of a finite total variation with the Fourier transform g(y) with the property g(y)=h(f(y)) for each y such that f(y) belongs to K.