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Proceedings of the London Mathematical Society Advance Access originally published online on November 30, 2007
Proceedings of the London Mathematical Society 2008 96(2):464-506; doi:10.1112/plms/pdm051
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© 2007 London Mathematical Society

Generalized coorbit theory, Banach frames, and the relation to {alpha}-modulation spaces

Stephan Dahlke

Universität Marburg
Fachbereich Mathematik und Informatik
Hans–Meerwein–Strasse
Lahnberge
D-35032 Marburg
Germany
dahlke@mathematik.uni-marburg.de

Massimo Fornasier

Program in Applied and Computational
Mathematics Princeton University
Fine Hall
Washington Road
08544-1000 Princeton, NJ
USA

Holger Rauhut

Universität Wien
Fakultät für Mathematik
NuHAG
Nordbergstrasse 15
A-1090 Wien
Austria
holger.rauhut@univie.ac.at

Gabriele Steidl

Universität Mannheim
Fakultät für Mathematik und Informatik
D7, 27
D-68131 Mannheim
Germany
steidl@math.uni-mannheim.de

Gerd Teschke

Konrad-Zuse-Zentrum für
Informationstechnik Berlin (ZIB)
Takustrasse 7
D-14195 Berlin-Dahlem
Germany
teschke@zib.de

Received 2 February 2006.

This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients, which has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness spaces that can be identified with {alpha}-modulation spaces.

The authors acknowledge the financial support provided through the European Union's Human Potential Programme, under contract HPRN-CT-2002-00285 (HASSIP), and through Deutsche Forschungsgemeinschaft (DFG), grants Da 360/4-2, Da 360/4-3, We 2602/2-1, Ma 1657/6-1, and Te 354/1-2. The second author also wants to thank the AG Numerik/Wavelet-Analysis Group, Philipps-Universität Marburg, and the ZeTeM, Universität Bremen, Germany, for the hospitality and the stimulating cooperations during the preparation of this work. He also acknowledges the support of the Individual Marie Curie Fellowship MEIF-CT-2004-501018 at NuHAG, Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Austria.

2000 Mathematics Subject Classification 42C15, 42C40, 46E15 (primary), 57S25, 46E35 (secondary).


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