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Coorbit Theory and Bergman Spaces

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Harmonic and Complex Analysis and its Applications

Part of the book series: Trends in Mathematics ((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).

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Notes

  1. 1.

    This option will become important for the L p-theory, with p> 1.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics, NuHAG, University Vienna, Nordbergstraße 15, 1090, Wien, Austria

    H. G. Feichtinger

  2. Institute of Mathematics and Informatics, University of Pécs, Ifjúság útja 6, 7634, Pécs, Hungary

    M. Pap

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  1. H. G. Feichtinger
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  2. M. Pap
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Correspondence to H. G. Feichtinger .

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Editors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Alexander Vasil'ev

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Feichtinger, H.G., Pap, M. (2014). Coorbit Theory and Bergman Spaces. In: Vasil'ev, A. (eds) Harmonic and Complex Analysis and its Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-01806-5_4

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