Frame expansions of test functions, tempered distributions, and ultradistributions

TL;DR

This paper explores frame expansions in Fréchet spaces, providing new results on series expansions of tempered distributions and ultradistributions using localized frames and sequence space coefficients.

Contribution

It introduces novel series expansion results for tempered distributions and ultradistributions via localized frames in Fréchet spaces.

Findings

New series expansion results for tempered distributions.

Extensions to ultradistributions and test functions.

Framework for localized frames in Fréchet spaces.

Abstract

The paper is devoted to frame expansions in Fr\'echet spaces. First we review some results which concern series expansions in general Fr\'echet spaces via Fr\'echet and General Fr\'echet frames. Then we present some new results on series expansions of tempered distributions and ultradistributions, and the corresponding test functions, via localized frames and coefficients in appropriate sequence spaces.