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On the Atomic Decomposition of Coorbit Spaces with Non-integrable Kernel

  • Stephan DahlkeEmail author
  • Filippo De Mari
  • Ernesto De Vito
  • Lukas Sawatzki
  • Gabriele Steidl
  • Gerd Teschke
  • Felix Voigtlaender
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This chapter is concerned with recent progress in the context of coorbit space theory. Based on a square-integrable group representation, the coorbit theory provides new families of associated smoothness spaces, where the smoothness of a function is measured by the decay of the associated voice transform. Moreover, by discretizing the representation, atomic decompositions and Banach frames can be constructed. Usually, the whole machinery works well if the associated reproducing kernel is integrable with respect to a weighted Haar measure on the group. In recent studies, it has turned out that to some extent coorbit spaces can still be established if this condition is violated. In this chapter, we clarify in which sense atomic decompositions and Banach frames for these generalized coorbit spaces can be obtained.

Keywords

Coorbit theory Group representations Smoothness spaces Atomic decompositions Banach frames 

Notes

Acknowledgements

F. Voigtlaender would like to thank Werner Ricker for helpful discussions regarding idempotent Fourier multipliers.

E. De Vito and F. De Mari are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Instituto Nazionale di Alta Matematica (INdAM).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Stephan Dahlke
    • 1
    Email author
  • Filippo De Mari
    • 2
  • Ernesto De Vito
    • 2
  • Lukas Sawatzki
    • 1
  • Gabriele Steidl
    • 3
  • Gerd Teschke
    • 4
  • Felix Voigtlaender
    • 5
  1. 1.FB12 Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  2. 2.Dipartimento di MatematicaUniversità di GenovaGenovaItaly
  3. 3.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany
  4. 4.Institute for Computational Mathematics in Science and Technology, Hochschule NeubrandenburgUniversity of Applied SciencesNeubrandenburgGermany
  5. 5.Katholische Universität Eichstätt-IngolstadtEichstättGermany

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