# Localization of Fr\'echet frames and expansion of generalized functions

## TL;DR

This paper extends the theory of Fréchet frames by analyzing matrix operators with decay properties under weaker assumptions, and applies localization techniques for frame expansions of distributions.

## Contribution

It introduces weaker assumptions for matrix operator continuity and extends localization results from Banach to Fréchet spaces, enabling frame expansions of distributions.

## Key findings

- Weaker assumptions still ensure operator continuity.
- Localization of Fréchet frames facilitates distribution expansions.
- Extension from Banach to Fréchet spaces achieved.

## Abstract

Matrix type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fr\'{e}chet spaces. Moreover, the localization of Fr\'{e}chet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.