Fourier transforms of bounded functions

Abstract

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a Hölder 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 ${\int }_{|t|>1}{e}^{-ist}f(t)dt/t^2$. 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 $x\mapsto \arctan (x/a)$.