Structured, compactly supported Banach frame decompositions of decomposition spaces

TL;DR

This paper develops a framework for constructing structured Banach frames and atomic decompositions for decomposition spaces, allowing for compact support and broad applicability including modulation, Besov, and shearlet spaces.

Contribution

It introduces verifiable conditions for prototype functions to form Banach frames or atomic decompositions in decomposition spaces, enabling analysis and synthesis sparsity equivalence.

Findings

Framework applies to various function spaces like modulation and Besov spaces.

Provides conditions for Banach frame and atomic decomposition construction.

Enables analysis sparsity to imply synthesis sparsity with respect to primal frames.

Abstract

$\newcommand{mc}[1]{\mathcal{#1}}$ $\newcommand{D}{\mc{D}(\mc{Q},L^p,\ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a space $\D$ is defined using a frequency covering $\mc{Q}=(Q_i)_{i\in I}$: If $(\varphi_i)_{i}$ is a suitable partition of unity subordinate to $\mc{Q}$, then $\Vert g\Vert_{\D}:=\left\Vert\left(\Vert\mc{F}^{-1}(\varphi_i\hat{g})\Vert_{L^p}\right)_{i}\right\Vert_{\ell_w^q}$. We assume $\mc{Q}=(T_iQ+b_i)_{i}$, with $T_i\in{\rm GL}(\Bbb{R}^d),b_i\in\Bbb{R}^d$. Given a prototype $\gamma$, we consider the system \[\Psi_{c}=(L_{c\cdot T_i^{-T}k}\gamma^{[i]})_{i\in I,k\in\Bbb{Z}^d}\text{ with }\gamma^{[i]}=|\det T_i|^{1/2}\, M_{b_i}(\gamma\circ T_i^T),\] with translation $L_x$ and modulation $M_{\xi}$. We provide verifiable conditions on $\gamma$ under which $\Psi_c$ forms a Banach frame or an atomic decomposition for $\D$, for small enough sampling density $c>0$. Our theory allows compactly supported prototypes and applies for arbitrary $p,q\in(0,\infty]$. Often, $\Psi_c$ is both a Banach frame and an atomic decomposition, so that analysis sparsity is equivalent to synthesis sparsity, i.e. the analysis coefficients $(\langle f,L_{c\cdot T_i^{-T}k}\gamma^{[i]}\rangle)_{i,k}$ lie in $\ell^p$ iff $f$ belongs to a certain decomposition space, iff $f=\sum_{i,k}c_k^{(i)}\cdot L_{c\cdot T_i^{-T}k}\gamma^{[i]}$ with $(c_k^{(i)})_{i,k}\in\ell^p$. This is convenient if only analysis sparsity is known to hold: Generally, this only yields synthesis sparsity w.r.t. the dual frame, about which often only little is known. But our theory yields synthesis sparsity w.r.t. the well-understood primal frame. In particular, our theory applies to $\alpha$-modulation spaces and inhom. Besov spaces. It also applies to shearlet frames, as we show in a companion paper.