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The Fourier transform in Lebesgue spaces

Erik Talvila

TL;DR

This work develops a robust distributional framework for the Fourier transform on all $L^p(\mathbb{R})$ spaces with $1\le p<\infty$ by representing the transform as the distributional derivative of a continuous primitive $\Psi_f$, and proving a sharp Hölder estimate $|\Psi_f(s)|\le C_q\|f\|_p|s|^{1/p}$. It defines Banach spaces $\mathcal{B}_p$ and $\mathcal{A}_p$ that are isometrically isomorphic to $L^p$, and realizes the Fourier transform as an isometric map $\mathcal{F}: L^p \to \mathcal{A}_p$ via $\hat{f}=\Psi_f'$, thereby enabling a norm-based inversion and exchange theory. The paper establishes an exchange formula and norm-inversion using canonical summability kernels, and develops an integration theory and convolution results within the $\mathcal{A}_p$ framework. This approach extends Fourier-analytic tools beyond the traditional $L^1$–$L^\infty$ and Hausdorff–Young regimes, providing precise growth, regularity, and inversion properties for $L^p$ functions in a distributional setting.

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 Hölder 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.

The Fourier transform in Lebesgue spaces

TL;DR

This work develops a robust distributional framework for the Fourier transform on all spaces with by representing the transform as the distributional derivative of a continuous primitive , and proving a sharp Hölder estimate . It defines Banach spaces and that are isometrically isomorphic to , and realizes the Fourier transform as an isometric map via , thereby enabling a norm-based inversion and exchange theory. The paper establishes an exchange formula and norm-inversion using canonical summability kernels, and develops an integration theory and convolution results within the framework. This approach extends Fourier-analytic tools beyond the traditional – and Hausdorff–Young regimes, providing precise growth, regularity, and inversion properties for functions in a distributional setting.

Abstract

For each () it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each a norm is defined so that the space Fourier transforms is isometrically isomorphic to . There is an exchange theorem and inversion in norm.
Paper Structure (7 sections, 9 theorems, 14 equations)

This paper contains 7 sections, 9 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Properties of $\Psi_{\!f}$
  3. Fourier transform and the Banach spaces ${\mathcal{B}}_p({\mathbb R})$ and ${\mathcal{A}}_p({\mathbb R})$
  4. Properties of the Fourier transform
  5. Integration in ${\mathcal{A}}_p({\mathbb R})$
  6. Exchange formula and inversion
  7. Convolution

Key Result

Theorem 2.1

Let $f\in L^{p}({\mathbb R})$ for some $1\leq p<\infty$. Define $\Psi_{\!f}(s)= \int^\infty_{-\infty}\left(\frac{1-e^{-ist}}{it}\right) f(t)\,dt$ for $s\in{\mathbb R}$. Let $1/p+1/q=1$. If $p=1$ then $q=\infty$. Let $C_q=4^{1/q}(\int_0^\infty\lvert\sin(t)/t\rvert^q\,dt)^{1/q}$, with $C_\infty=1$. (a

Theorems & Definitions (31)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Theorem 3.1
  • proof
  • Definition 3.2
  • ...and 21 more