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From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces


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Feichtinger, Hans Georg ; Voigtlaender, Felix:
From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces.
In: Cwikel, Michael ; Milman, Mario (Hrsg.): Functional analysis, harmonic analysis, and image processing : a collection of papers in honor of Björn Jawerth. - Providence, Rhode Island : American Mathematical Society, 2017. - S. 185-216. - (Contemporary mathematics ; 693)
ISBN 978-1-4704-2836-5 ; 978-1-4704-4166-1


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This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces.
Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution under consideration. As a concrete example, one might think of the wavelet transform, leading to the theory of Besov-Triebel-Lizorkin spaces.
Decomposition spaces, on the other hand, are defined using certain decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces.
Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces. In this way, optimal inclusion results and invariance properties for a variety of smoothness spaces can be established. We will present an outline of these connections and comment on the basic results arising in this context.

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Publikationsform:Aufsatz in einem Buch
Sprache des Eintrags:Englisch
Institutionen der Universität:Mathematisch-Geographische Fakultät > Mathematik > Lehrstuhl für Reliable Machine Learning
Begutachteter Aufsatz:Ja
Titel an der KU entstanden:Nein
Eingestellt am: 01. Apr 2022 14:20
Letzte Änderung: 05. Apr 2022 12:21
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