Pointwise characterizations in generalized function algebras

TL;DR

This paper introduces a new algebra of generalized functions on generalized points, characterizes subalgebras via pointwise properties, and clarifies the relationship between different notions of regularity in generalized function algebras.

Contribution

It defines a new algebra of Colombeau generalized functions on generalized points and characterizes subalgebras and regularity through pointwise properties, clarifying previous misconceptions.

Findings

Characterization of al S-regular generalized functions by pointwise Fourier transform properties.

Equality in generalized tempered distributions characterized by pointwise Fourier transform properties.

Proved that al G^\u221e(\u03a9) equals al G^\u221e() for open subsets  of b^d.

Abstract

We define the algebra of Colombeau generalized functions on the space of generalized points of {\mathbb R}^d which naturally contains the tempered generalized functions. The subalgebra of \mathscr{S}-regular generalized functions of this algebra is characterized by a pointwise property of the generalized functions and their Fourier transforms. We also characterize the equality in the sense of generalized tempered distributions for certain elements of this algebra (namely those with so-called slow scale support) by means of a pointwise property of their Fourier transforms. Further, we show that (contrary to what has been claimed in the literature) for an open subset \Omega of {\mathbb R}^d, the algebra of pointwise regular generalized functions \dot{\mathcal G}^\infty(\Omega) equals {\mathcal G}^\infty(\Omega) and give several characterizations of pointwise {\mathcal G}^\infty-regular generalized functions.