Structured, compactly supported Banach frame decompositions of decomposition spaces
Felix Voigtlaender

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 is defined using a frequency covering : If is a suitable partition of unity subordinate to , then . We assume , with . Given a prototype , 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 and modulation . We provide verifiable conditions on under…
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Taxonomy
TopicsMathematical Analysis and Transform Methods
