Rapidly converging approximations and regularity theory

TL;DR

This paper develops a new framework for distributions on compact manifolds using smoothing operators, introducing an algebra of generalized functions with group invariance and regularity properties linked to tameness in graded Fréchet spaces.

Contribution

It constructs an algebra of generalized functions for distributions on manifolds, embedding distributions in a diffeomorphism-equivariant way and analyzing regularity via tameness in graded Fréchet spaces.

Findings

Established a diffeomorphism-equivariant embedding of distributions.

Constructed an algebra of generalized functions invariant under group actions.

Linked regularity in this framework to Colombeau algebra regularity.

Abstract

We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $\Psi^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frech\'et spaces. This notion of regularity implies the regularity in Colombeau algebras in the $\maG^{\infty}$ sense.