## Structured, compactly supported Banach frame decompositions of decomposition spaces

### Volume 575 / 2022

#### Abstract

This paper presents a framework for constructing structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a decomposition space $\def\DecompSp#1#2#3#4{{\mathcal{D}({#1},L_{#4}^{#2},{#3})}}\DecompSp{\mathcal Q}p{\ell_{w}^{q}}{}$ is defined using a frequency covering $\mathcal Q = (Q_{i})_{i\in I}$ and a suitable weight $w = (w_{i} )_{i\in I}$: If $ (\varphi_{i} )_{i\in I}$ is a suitable partition of unity subordinate to $\mathcal Q$, then the decomposition space norm is given by $\def\DecompSp#1#2#3#4{{\mathcal{D}({#1},L_{#4}^{#2},{#3})}} \Vert g \Vert_{\DecompSp{\mathcal Q}p{\ell_{w}^{q}}{}} =\Vert ( w_{i} \cdot \Vert \mathcal{F} ^{-1} ( \varphi_{i} \cdot \widehat{g} \, ) \Vert_{L^{p}} )_{i\in I} \Vert_{\ell^{q}} . $ We assume $\mathcal Q= (T_{i}Q+b_{i} )_{i\in I}$, with $T_{i}\in{\rm GL} (\mathbb R^{d} )$ and $b_{i}\in\mathbb R^{d}$.

Given a prototype $\gamma$, we provide characterizations of the spaces
$\def\DecompSp#1#2#3#4{{\mathcal{D}({#1},L_{#4}^{#2},{#3})}}\DecompSp{\mathcal Q}p{\ell_{w}^{q}}{}$ using a system of the form
\[
\Psi_{c}= (L_{c\cdot T_{i}^{-T}k}\,\gamma^{ [i ]} )\quad\text{where}\quad\gamma^{ [i ]}= |\!\det T_{i} |^{1/2}\cdot M_{b_{i}} (\gamma\circ{T_{i}^{T}} ),
\]
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 $\def\DecompSp#1#2#3#4{{\mathcal{D}({#1},L_{#4}^{#2},{#3})}}\DecompSp{\mathcal Q}p{\ell_{w}^{q}}{}$,
for small enough sampling density $c \gt 0$. Our theory allows compactly
supported prototypes and applies to arbitrary $p,q\in (0,\infty ]$.
In many cases, $\Psi_{c}$ forms both a Banach frame and an atomic
decomposition, so that analysis sparsity is equivalent to synthesis
sparsity, that is, the analysis coefficients $ ( \langle f,L_{c\cdot T_{i}^{-T}k}\,\gamma^{ [i ]} \rangle )_{i\in I,\,k\in\mathbb Z^{d}}$
lie in $\ell^{p}$ if and only if $f$ belongs to a certain decomposition
space, if and only if $f=\sum_{i,k}c_{k}^{ (i )}\,L_{c\cdot T_{i}^{-T}k}\,\gamma^{ [i ]}$
with $ (\smash{c_{k}^{ (i )}} )_{i\in I,\,k\in\mathbb Z^{d}}\in\ell^{p}$.
This is convenient for example when only analysis sparsity is known
to hold: Generally, this only yields synthesis sparsity *with
respect to the dual frame*, about which often only little is known.
In contrast, our theory yields synthesis sparsity with respect to
the well-understood primal frame. In particular, our theory applies
to $\alpha$-modulation spaces and inhomogeneous Besov spaces. It
also applies to cone-adapted shearlet frames, as we show in the companion
paper *Analysis vs. synthesis sparsity for $\alpha$-shearlets*
[arXiv:1702.03559 (2017)].