Some properties of Fourier integrals

TL;DR

This paper explores new properties of Fourier transform algebras, providing criteria for membership, and investigates the Hilbert operator's isometric properties and integral representations of Carleman transforms.

Contribution

It introduces new properties of Fourier transform algebras, criteria for membership, and analyzes the Hilbert operator's isometric nature in these spaces.

Findings

Criteria for membership in Fourier transform algebras are established.

The Hilbert operator is shown to be a bijective isometry in certain Banach spaces.

Integral representations of Carleman transforms for specific measure classes are derived.

Abstract

Let F(R^n) be the algebra of Fourier transforms of functions from L_1(R^n), K(R^n) be the algebra of Fourier transforms of bounded complex Borel measures in R^n and W be Wiener algebra of continuous 2pi-periodic functions with absolutely convergent Fourier series. New properties of functions from these algebras are obtained. Some conditions which determine membership of f in F(R) are given. For many elementary functions f the problem of belonging f to F(R) can be resolved easily using these conditions. We prove that the Hilbert operator is a bijective isometric operator in the Banach spaces W_0, F(R), K(R)-A_1 (A_1 is the one-dimension space of constant functions). We also consider the classes M_k, which are similar to the Bochner classes F_k, and obtain integral representation of the Carleman transform of measures of M_k by integrals of some specific form.