On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution

TL;DR

This paper develops a convolution theory for quasianalytic Gelfand-Shilov ultradistributions, introduces ultrapolynomials for the parametrix method, and explores new properties of quasianalytic function spaces including Fourier hyperfunctions.

Contribution

It introduces a convolution framework and ultrapolynomials for quasianalytic Gelfand-Shilov spaces, advancing the understanding of their topological and structural properties.

Findings

Established a convolution theory for quasianalytic ultradistributions.

Constructed ultrapolynomials as a basis for the parametrix method.

Applied results to Fourier hyperfunctions and ultra-hyperfunctions.

Abstract

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.