Coorbit Theory and Bergman Spaces

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


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).


Atomic decomposition Bergman spaces Coorbit spaces 


  1. 1.
    J. Arazy, Some aspects of the minimal, Möbius-invariant space of analytic functions on the unit disc., Interpolation spaces and allied topics in analysis, Proc. Conf., Lund/Swed. 1983, Lect. Notes Math. 1070, 24–44 (1984)Google Scholar
  2. 2.
    J. Arazy, S. Fisher, J. Peetre, Möbius invariant function spaces., J. Reine Angew. Math. 363, 110–145 (1985)Google Scholar
  3. 3.
    P. Boggiatto, E. Cordero, K. Gröchenig, Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integr. Equat. Oper. Theory 48(4), 427–442 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    J.G. Christensen, G. Olafson, Examples of coorbit spaces for dual pairs. Acta Appl. Math. 107(1–2), 25–48 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J.G. Christensen, G. Olafsson, Coorbit spaces for dual pairs. Appl. Comput. Harmon. Anal. 31(2), 303–324 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J.G. Christensen, A. Mayeli, G. Olafsson, Coorbit description and atomic decomposition of Besov spaces. Arxiv preprint arXiv:1110.6676 (2011)Google Scholar
  7. 7.
    J.G. Christensen, A. Mayeli, G. Olafsson, Coorbit description and atomic decomposition of Besov spaces. Numer. Funct. Anal. Opt. 33(7–9), 847–871 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. Cordero, L. Rodino, Wick calculus: a time-frequency approach. Osaka J. Math. 42(1), 43–63 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Dahlke, G. Steidl, G. Teschke, Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere. Adv. Comput. Math. 21(1–2), 147–180 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Dahlke, G. Steidl, G. Teschke, Weighted coorbit spaces and Banach frames on homogeneous spaces. J. Fourier Anal. Appl. 10(5), 507–539 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. Dahlke, G. Teschke, K. Stingl, Coorbit theory, multi-α-modulation frames, and the concept of joint sparsity for medical multichannel data analysis, 2008,
  12. 12.
    S. Dahlke, M. Fornasier, H. Rauhut, G. Steidl, G. Teschke, Generalized coorbit theory, Banach frames, and the relation to alpha-modulation spaces. Proc. Lond. Math. Soc. 96(2), 464–506 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. Dahlke, G. Kutyniok, G. Steidl, G. Teschke, Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harmon. Anal. 27(2), 195–214 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Dahlke, G. Steidl, G. Teschke, Shearlet coorbit spaces: Compactly supported analyzing shearlets, traces and embeddings. J. Fourier Anal. Appl. 17(6), 1232–1255 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. Dahlke, G. Steidl, G. Teschke, Multivariate shearlet transform, shearlet coorbit spaces and their structural properties,
  16. 16.
    S. Dahlke, S. Häuser, G. Teschke, Coorbit space theory for the Toeplitz shearlet transform, Int. J. Wavelets Multiresolut. Inf. Process. 10(04), p.1250037, 13 p. (2012)Google Scholar
  17. 17.
    P. Duren, A. Schuster, Bergman Spaces, vol. 100 of Mathematical Surveys and Monographs. (Amer. Math. Soc. (AMS), Providence, RI, 2004)Google Scholar
  18. 18.
    H.G. Feichtinger, On a new Segal algebra. Monatsh. Math. 92, 269–289 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    H.G. Feichtinger, K. Gröchenig, A unified approach to atomic decompositions via integrable group representations. Lect. Note Math. 1302, 52–73 (1988)CrossRefGoogle Scholar
  20. 20.
    H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86(2), 307–340 (1989)CrossRefzbMATHGoogle Scholar
  21. 21.
    H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math. 108(2–3), 129–148 (1989)CrossRefzbMATHGoogle Scholar
  22. 22.
    M. Fornasier, Nota on the coorbit theory of alpha- modulation spaces, 2005Google Scholar
  23. 23.
    M. Fornasier, H. Rauhut, Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl. 11(3), 245–287 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M. Fornasier, Banach frames for α-modulation spaces. Appl. Comput. Harmon. Anal. 22(2), 157–175 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    K. Gröchenig, Describing functions: atomic decompositions versus frames. Monatsh. Math. 112(3), 1–41 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    K. Gröchenig, M. Piotrowski, Molecules in coorbit spaces and boundedness of operators. Studia Math. 192(1), 61–77 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    K. Gröchenig, J. Toft, Isomorphism properties of Toeplitz operators in time-frequency analysis. J. d’Analyse 114(1), 255–283 (2011)CrossRefzbMATHGoogle Scholar
  28. 28.
    K. Gröchenig, J. Toft, The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces. Trans. Am. Math. Soc. 23 (2012)Google Scholar
  29. 29.
    C. Heil, D.F. Walnut, Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces (Springer, New York, 2000)CrossRefzbMATHGoogle Scholar
  31. 31.
    G. Kutyniok, et al. (eds.) Shearlets. Multiscale Analysis for Multivariate Data (Birkhäuser, Boston, MA). Appl. Numer. Harmonic Anal. 105–144 (2012)Google Scholar
  32. 32.
    M. Mantoiu, Quantization rules, Hilbert algebras and coorbit spaces for families of bounded operators I. The Abstract Theory. Arxiv preprint arXiv:1203.6347 (2012)Google Scholar
  33. 33.
    M. Pap, F. Schipp, The voice transform on the Blaschke group III. Publ. Math. Debrecen 75(1–2), 263–283 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    M. Pap, The voice transform generated by a representation of the Blaschke group on the weighted Berman spaces. Ann. Univ. Sci. Budapest. Sect. Comp. 33, 321–342 (2010)Google Scholar
  35. 35.
    M. Pap, Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces. J. Math. Anal. Appl. 389(1), 340–350 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    A. Perälä, J. Taskinen, J. Virtanen, New results and open problems on Toeplitz operators in Bergman spaces. New York J. Math. 17a, 147–164 (2011)Google Scholar
  37. 37.
    H. Rauhut, Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups. Appl. Comput. Harmon. Anal. 18(1), 94–122 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    H. Rauhut, Coorbit space theory for quasi-Banach spaces. Studia Math. 180(3), 237–253 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    H. Rauhut, T. Ullrich, Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type. J. Funct. Anal. 11, 3299–3362 (2011)MathSciNetCrossRefGoogle Scholar
  40. 40.
    J.L. Romero, Characterization of coorbit spaces with phase-space covers. J. Funct. Anal. 262(1), 59–93 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    F. Schipp, W. Wade, Transforms on Normed Fields, Leaflets in Mathematics (Janus Pannonius University, Pecs, 1995)Google Scholar
  42. 42.
    J. Toft, P. Boggiatto, Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces. Adv. Math. 217(1), 305–333 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, vol. 226 of Graduate Texts in Mathematics (Springer, New York, NY, 2005)Google Scholar

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

Personalised recommendations