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Structured, compactly supported Banach frame decompositions of decomposition spaces
$\newcommand{mc}[1]{\mathcal{#1}}$ $\newcommand{D}{\mc{D}(\mc{Q},L^p,\ell_w^q)}$ 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 $(φ_i)_{i}$ is a suitable partition of unity subordinate to $\mc{Q}$, then $\Vert g\Vert_{\D}:=\left\Vert\left(\Vert\mc{F}^{-1}(φ_i\hat{g})\Vert_{L^p}\right)_{i}\right\Vert_{\ell_w^q}$. 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 $γ$, we consider the system \[Ψ_{c}=(L_{c\cdot T_i^{-T}k}γ^{[i]})_{i\in I,k\in\Bbb{Z}^d}\text{ with }γ^{[i]}=|\det T_i|^{1/2}\, M_{b_i}(γ\circ T_i^T),\] with translation $L_x$ and modulation $M_ξ$. We provide verifiable conditions on $γ$ under which $Ψ_c$ 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, $Ψ_c$ is both a Banach frame and an atomic decomposition, so that analysis sparsity is equivalent to synthesis sparsity, i.e. the analysis coefficients $(\langle f,L_{c\cdot T_i^{-T}k}γ^{[i]}\rangle)_{i,k}$ lie in $\ell^p$ iff $f$ belongs to a certain decomposition space, iff $f=\sum_{i,k}c_k^{(i)}\cdot L_{c\cdot T_i^{-T}k}γ^{[i]}$ 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 $α$-modulation spaces and inhom. Besov spaces. It also applies to shearlet frames, as we show in a companion paper.
