Fourier transforms of bounded functions

TL;DR

This paper characterizes the Fourier transform of bounded functions as second derivatives of Hölder continuous functions, providing new integral representations, isometric isomorphisms, and explicit transforms for specific functions.

Contribution

It introduces a novel representation of Fourier transforms of bounded functions as second derivatives of Hölder functions and establishes isometric isomorphism with $L^ Infty$, including explicit formulas for certain functions.

Findings

Fourier transform of bounded functions as second derivatives of Hölder functions

Isometric isomorphism between Fourier transforms and $L^ Infty$

Explicit Fourier transforms for functions like $rac{ ext{cos}^m(a/x)}{x}$, $x ext{sin}(a/x)$, and $ ext{arctan}(x/a)

Abstract

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1 e^{-ist}f(t)\,dt$ and the second distributional derivative of the integral $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. The space of such Fourier transforms is isometrically isomorphic to $L^\infty(\mathbb{R})$. There is an exchange theorem, inversion and convolution results. The Fourier transform of the functions $x\mapsto\cos^m(a/x)$ for each natural number $m$ are computed. Also for $x\mapsto x\sin(a/x)$ and $x\mapsto\arctan(x/a)$.