Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $(\varphi_i)_{i}$ 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.

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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an