# Invertibility of frame operators on Besov-type decomposition spaces

## TL;DR

This paper extends the Walnut-Daubechies criterion to establish the invertibility of frame operators on Besov-type spaces, enabling broader applications of $L^2$ frame expansions to various function spaces.

## Contribution

It generalizes the invertibility criterion for frame operators to Besov-type spaces, improving understanding of atomic decompositions and extending $L^2$ frame expansions.

## Key findings

- Frame operators are invertible on Besov-type spaces under the extended criterion.
- $L^2$ frame expansions extend to many other function spaces beyond $L^2$.
- Operations like analysis, thresholding, and synthesis are bounded on these spaces.

## Abstract

We derive an extension of the Walnut-Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that $L^2$ frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the $L^2$ canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces.