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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)].