A Fourier transform for all generalized functions

TL;DR

This paper introduces a hyperfinite Fourier transform within non-Archimedean generalized function spaces, extending classical Fourier analysis to broader contexts including distributions and Colombeau functions.

Contribution

It defines a new Fourier transform using non-Archimedean analysis that applies to all generalized smooth functions, overcoming previous limitations and preserving key properties.

Findings

Generalizes classical Fourier transform properties

Applies to all Schwartz distributions and Colombeau functions

Enables non-tempered solutions to differential equations

Abstract

Using the existence of infinite numbers $k$ in the non-Archimedean ring of Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by considering integration extended to $[-k,k]^{n}$ instead of $(-\infty,\infty)^{n}$. In order to realize this idea, the space of generalized functions we consider is that of generalized smooth functions (GSF), an extension of classical distribution theory sharing many nonlinear properties with ordinary smooth functions, like the closure with respect to composition, a good integration theory, and several classical theorems of calculus. Even if the final transform depends on $k$, we obtain a new notion that applies to all GSF, in particular to all Schwartz's distributions and to all Colombeau generalized functions, without growth restrictions. We prove that this FT generalizes several classical properties of the ordinary FT, and in this way we also overcome the difficulties of FT in Colombeau's settings. Differences in some formulas, such as in the transform of derivatives, reveal to be meaningful since allow to obtain also non-tempered global unique solutions of differential equations.