Some extensions to the functional analytic approach to Colombeau algebra

TL;DR

This paper extends the functional analytic framework for Colombeau algebras to include tempered generalized functions, defining a Fourier transform with a strict inversion theorem that aligns with classical Fourier analysis.

Contribution

It introduces a new definition of Fourier transform for nonlinear generalized functions within Colombeau algebras, maintaining classical properties and enabling inversion.

Findings

Fourier transform for nonlinear generalized functions is rigorously defined.

The transform agrees with classical Fourier transform on tempered distributions.

A strict inversion theorem for the Fourier transform is established.

Abstract

We extend the functional analytic approach to Colombeau-type spaces of nonlinear generalized functions in order to study algebras of tempered generalized functions. We obtain a definition of Fourier transform of nonlinear generalized functions which has a strict inversion theorem, agrees with the classical Fourier transform for tempered distributions and preserves well-known classical properties.