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Coorbit Theory and Bergman Spaces

  • H. G. FeichtingerEmail author
  • M. Pap
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Coorbit theory arose as an attempt to describe in a unified fashion the properties of the continuous wavelet transform and the STFT (Short-time Fourier transform) by taking a group theoretical viewpoint. As a consequence H.G. Feichtinger and K.H. Gröchenig have established a rather general approach to atomic decomposition for families of Banach spaces (of functions, distributions, analytic functions, etc.) through integrable group representations (see Feichtinger and Gröchenig (Lect. Notes Math. 1302:52–73, 1988; J. Funct. Anal. 86(2):307–340, 1989; Monatsh. Math. 108(2–3):129–148, 1989), Gröchenig (Monatsh. Math. 112(3):1–41, 1991)), now known as coorbit theory.They gave also examples for the abstract theory and until now this approach gives new insights on atomic decompositions, even for cases where concrete examples can be obtained by other methods. Due to the flexibility of this theory the class of possible atoms is much larger than it was supposed to be in concrete cases. It is a remarkable fact that almost all classical function spaces in real and complex variable theory occur naturally as coorbit spaces related to certain integrable representations.In the present paper we present an overview of the general theory and applications for the case of the weighted Bergman spaces over the unit disc, indicating the benefits of the group theoretic perspective (more flexibility, at least at a qualitative level, more general atoms).

Keywords

Atomic decomposition Bergman spaces Coorbit spaces 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Faculty of Mathematics, NuHAGUniversity ViennaWienAustria
  2. 2.Institute of Mathematics and InformaticsUniversity of PécsPécsHungary

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