# Design and properties of wave packet smoothness spaces

## TL;DR

This paper introduces wave packet smoothness spaces, a new family of quasi-Banach spaces characterized by sparsity in frame expansions, and explores their structure, embeddings, and relations to classical function spaces.

## Contribution

The paper defines wave packet smoothness spaces, constructs Banach frames and atomic decompositions, and analyzes their embeddings with classical function spaces.

## Key findings

- Wave packet smoothness spaces include spaces characterized by sparsity in Gabor and wave atom expansions.
- Banach frames and atomic decompositions are constructed for these spaces.
- Embeddings between wave packet smoothness spaces and classical spaces like Besov and Sobolev are established.

## Abstract

We introduce a family of quasi-Banach spaces - which we call wave packet smoothness spaces - that includes those function spaces which can be characterised by the sparsity of their expansions in Gabor frames, wave atoms, and many other frame constructions. We construct Banach frames for and atomic decompositions of the wave packet smoothness spaces and study their embeddings in each other and in a few more classical function spaces such as Besov and Sobolev spaces.