Generalized function algebras containing spaces of periodic ultradistributions

TL;DR

This paper develops generalized differential algebras that embed spaces of periodic ultradistributions, including hyperfunctions, ensuring consistency with pointwise multiplication and characterizing regularity within the algebra.

Contribution

It constructs optimal embeddings of periodic ultradistributions into differential algebras, preserving multiplication and defining a notion of regularity.

Findings

Embedded hyperfunctions into a differential algebra.

Proved the optimality of embeddings via Schwartz impossibility results.

Characterized regular ultradistributions as ultradifferentiable functions.

Abstract

We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. In particular, we embed the space of hyperfunctions on the unit circle into a differential algebra in such a way that the multiplication of real analytic functions on the unit circle coincides with their pointwise multiplication. Furthermore, we introduce a notion of regularity in our newly defined algebras and show that an embedded ultradistribution is regular if and only if it is an ultradifferentiable function.