A convenient notion of compact set for generalized functions

TL;DR

This paper introduces the concept of functionally compact sets within Colombeau's generalized functions, creating spaces analogous to classical test functions and establishing their topological and functional properties.

Contribution

It defines functionally compact sets for generalized functions and develops associated spaces, bridging properties of smooth functions on compact sets with generalized function theory.

Findings

Defined functionally compact sets in Colombeau theory

Constructed spaces of compactly supported generalized smooth functions

Established topological and functional analytic foundations for these spaces

Abstract

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional analytic foundations of these spaces.