Fourier transforms of bounded functions
Erik Talvila
TL;DR
The paper defines the Fourier transform of bounded measurable functions as the second distributional derivative of a Hölder continuous primitive, expressing hat{f} as hat{f1} + hat{f2} with hat{f2} = −Ψ_f'' where Ψ_f(s)=∫_{|t|>1} e^{-ist} f(t) dt / t^2. It develops a robust Banach-space framework, showing isometric isomorphisms between L^ty(ℝ), the transform space A_ty(ℝ), and the primitive space B_ty(ℝ), and proves an exchange theorem valid for test functions with bounded-variation derivatives. An inversion theory via summability kernels provides both norm- and pointwise convergence results, while a convolution theory extends classical identities to this distributional setting. The paper also computes the Fourier transforms of several bounded, singular examples (e.g., cos^m(a/x), x sin(a/x), arctan(x/a)), illustrating delta contributions and distributional limits. Overall, it broadens Fourier analysis of bounded signals by integrating distributional derivatives, Hölder regularity, and weak inversion/convolution results, enabling practical analysis beyond traditional L^1/L^2 frameworks.
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 $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. 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 $x\mapsto\arctan(x/a)$.