The Fourier transform in Lebesgue spaces

TL;DR

This paper characterizes the Fourier transform on Lebesgue spaces as a distributional derivative of a Hölder continuous function, establishing isometric isomorphisms and inversion properties in this context.

Contribution

It introduces a new perspective on the Fourier transform in Lebesgue spaces, defining a norm that makes the transform an isometric isomorphism and proving an exchange theorem and inversion in norm.

Findings

Fourier transform as distributional derivative of Hölder continuous functions

Isometric isomorphism between Fourier transform space and L^p spaces

Established exchange theorem and inversion in norm

Abstract

For each $f\in L^p({\mathbb R)}$ ($1\leq p<\infty$) it is shown that the Fourier transform is the distributional derivative of a H\"older continuous function. For each $p$ a norm is defined so that the space Fourier transforms is isometrically isomorphic to $L^p({\mathbb R)}$. There is an exchange theorem and inversion in norm.