Function Spaces of Polyanalytic Functions

  • Luis Daniel AbreuEmail author
  • Hans G. Feichtinger
Part of the Trends in Mathematics book series (TM)


This article is meant as both an introduction and a review of some of the recent developments on Fock and Bergman spaces of polyanalytic functions. The study of polyanalytic functions is a classic topic in complex analysis. However, thanks to the interdisciplinary transference of knowledge promoted within the activities of HCAA network it has benefited from a cross-fertilization with ideas from signal analysis, quantum physics, and random matrices. We provide a brief introduction to those ideas and describe some of the results of the mentioned cross-fertilization. The departure point of our investigations is a thought experiment related to a classical problem of multiplexing of signals, in order words, how to send several signals simultaneously using a single channel.


Gabor frames Landau levels Polyanalytic spaces 



The authors wish to thank Radu Frunza for sharing his MATLAB-code and to Franz Luef, José Luis Romero, Karlheinz Gröchenig, Luis V. Pessoa, and Tomasz Hrycak for interesting discussions and comments on early versions of these notes.

L.D. Abreu was supported by CMUC and FCT project PTDC/MAT/114394/2009 through COMPETE/FEDER and by Austria Science Foundation (FWF) projects “‘Frames and Harmonic Analysis”’ and START-project FLAME.


  1. 1.
    L.D. Abreu, Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Appl. Comp. Harm. Anal. 29, 287–302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    L.D. Abreu, On the structure of Gabor and super Gabor spaces. Monatsh. Math. 161, 237–253 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L.D. Abreu, K. Gröchenig, Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group. Appl. Anal. 91, 1981–1997 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L.D. Abreu, Wavelet frames with Laguerre functions. C. R. Acad. Sci. Paris Ser. I 349, 255–258 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    L.D. Abreu, Super-wavelets versus poly-Bergman spaces. Int. Eq. Op. Theor. 73, 177–193 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L.D. Abreu, Wavelet (super)frames with Laguerre functions, ongoing workGoogle Scholar
  7. 7.
    L.D. Abreu, M. de Gosson, Displaced coherent states and true polyanalytic Fock spaces, ongoing workGoogle Scholar
  8. 8.
    L.D. Abreu, N. Faustino, On toeplitz operators and localization operators. Proc. Am. Math. Soc. (to appear)Google Scholar
  9. 9.
    M.L. Agranovsky, Characterization of polyanalytic functions by meromorphic extensions into chains of circles. J. d’Analyse Math. 113, 305–329 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S.T. Ali, J.P. Antoine, J.P. Gazeau, Coherent States and Their Generalizations (Springer, Berlin, 2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Y. Ameur, H. Hedenmalm, N. Makarov, Berezin transform in polynomial Bergman spaces. Comm. Pure Appl. Math. 63, 1533–1584 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. Ascensi, J. Bruna, Model space results for the Gabor and Wavelet transforms. IEEE Trans. Inform. Theory 55, 2250–2259 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    N. Askour, A. Intissar, Z. Mouayn, Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants. C. R. Acad. Sci. Paris Sér. I Math. 325, 707–712 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    R. Balan, Multiplexing of signals using superframes, In SPIE Wavelets applications, vol. 4119 of Signal and Image processing XIII, pp. 118–129 (2000)Google Scholar
  15. 15.
    M.B. Balk, Polyanalytic Functions (Akad. Verlag, Berlin, 1991)zbMATHGoogle Scholar
  16. 16.
    H. Begehr, G.N. Hile, A hierarchy of integral operators. Rocky Mountain J. Math. 27, 669–706 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    H. Begehr, Orthogonal decompositions of the function space \({L}^{2}(\overline{D}, \mathbb{C})\). J. Reine Angew. Math. 549, 191–219 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    J. Ben Hough, M. Krishnapur, Y. Peres, B. Virág, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, University Lecture Series, vol. 51, x+154 (American Mathematical Society, Providence, RI, 2009)Google Scholar
  19. 19.
    A.J. Bracken, P. Watson, The quantum state vector in phase space and Gabor’s windowed Fourier transform. J. Phys. A 43, art. no. 395304 (2010)Google Scholar
  20. 20.
    S. Brekke, K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space. III. Math. Scand. 73, 112–126 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21.
    K. Bringmann, K. Ono, Dyson’s ranks and Maass forms. Ann. Math. 171, 419–449 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. Comtet, On the Landau levels on the hyperbolic plane. Ann. Phys. 173, 185–209 (1987)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Z. Cuckovic, T. Le, Toeplitz operators on Bergman spaces of polyanalytic functions. Bull. Lond. Math. Soc. 44(5), 961–973 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    I. Daubechies, “Ten Lectures On Wavelets”, CBMS-NSF Regional conference series in applied mathematics (1992)Google Scholar
  25. 25.
    I. Daubechies, J.R. Klauder, Quantum-mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II. J. Math. Phys. 26(9), 2239–2256 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    M. de Gosson, Spectral properties of a class of generalized Landau operators. Comm. Partial Differ. Equat. 33(10–12), 2096–2104 (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    M. de Gosson, F. Luef, Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization. J. Pseudo-Differ. Oper. Appl. 1(1), 3–34 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    M. Dörfler, J.L. Romero, Frames adapted to a phase-space cover, preprint arXiv:1207.5383 (2012)Google Scholar
  29. 29.
    P. Duren, A. Schuster, “Bergman Spaces”, Mathematical Surveys and Monographs, vol. 100 (American Mathematical Society, Providence, RI, 2004)Google Scholar
  30. 30.
    P. Duren, E.A. Gallardo-Gutiérrez, A. Montes-Rodríguez, A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces. Bull. Lond. Math. Soc. 39, 459–466 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73–90 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    H.G. Feichtinger, Modulation spaces on locally compact abelian groups. In: Proceedings of “International Conference on Wavelets and Applications” 2002, pp. 99–140, Chennai, India, 2003. Updated version of a technical report, University of Vienna, 1983Google Scholar
  33. 33.
    H.G. Feichtinger, On a new Segal algebra. Monatsh. Math. 92(4), 269–289 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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
  35. 35.
    H.G. Feichtinger, K. Gröchenig, A unified approach to atomic decompositions via integrable group representations. In: Proc. Function Spaces and Applications, Conf. Lund, 1986. Lect. Notes Math., vol. 1302 (Springer, New York, 1988), p. 5273Google Scholar
  36. 36.
    H.G. Feichtinger, M. Pap, Connection between the coorbit theory and the theory of Bergman spaces, this volume (2013)Google Scholar
  37. 37.
    R.P. Feynman, An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108–128 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    H. Führ, Simultaneous estimates for vector-valued Gabor frames of Hermite functions. Adv. Comput. Math. 29(4), 357–373 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    I. Gertner, G.A. Geri, Image representation using Hermite functions. Biol. Cybernetics 71(2), 147–151 (1994)CrossRefzbMATHGoogle Scholar
  40. 40.
    J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    K. Gröchenig, Describing functions: Atomic decompositions versus frames. Monatsh. Math. 112, 1–42 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    K. Gröchenig, “Foundations of Time-Frequency Analysis” (Birkhäuser, Boston, 2001)CrossRefzbMATHGoogle Scholar
  43. 43.
    K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10, 105–132 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    K. Gröchenig, Y. Lyubarskii, Gabor frames with Hermite functions. C. R. Acad. Sci. Paris Ser. I 344, 157–162 (2007)CrossRefzbMATHGoogle Scholar
  45. 45.
    K. Gröchenig, Y. Lyubarskii, Gabor (super)frames with Hermite functions. Math. Ann. 345, 267–286 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    K. Gröchenig, J. Stoeckler, Gabor frames and totally positive functions. Duke Math. J. 162(6), 1003–1031 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    A. Haimi, H. Hedenmalm, The polyanalytic Ginibre ensembles. Preprint arXiv:1106.2975 (2012)Google Scholar
  48. 48.
    A. Haimi, H. Hedenmalm, Asymptotic expansions of polyanalytic Bergman kernels. Preprint arXiv:1303.0720 (2013)Google Scholar
  49. 49.
    D. Han, D.R. Larson, Frames, bases and group representations. Mem. Am. Math. Soc. 147, 697 (2000)MathSciNetGoogle Scholar
  50. 50.
    C. Heil, History and evolution of the Density Theorem for Gabor frames. J. Fourier Anal. Appl. 13, 113–166 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    H. Hedenmalm, B. Korenblum, K. Zhu, The Theory of Bergman Spaces (Springer, New York, 2000). ISBN 978-0-387-98791-0CrossRefGoogle Scholar
  52. 52.
    O. Hutnik, A note on wavelet subspaces. Monatsh. Math. 160, 59–72 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    O. Hutník, M. Hutníková, An alternative description of Gabor spaces and Gabor-Toeplitz operators. Rep. Math. Phys. 66(2), 237–250 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    A. Jaffe, F. Quinn, Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics. Bull. Am. Math. Soc. (N.S.) 29, 1–13 (1993)Google Scholar
  55. 55.
    G.V. Kolossov, Sur les problêms d’élasticité a deux dimensions. C. R. Acad. Sci. 146, 522–525 (1908)Google Scholar
  56. 56.
    A.D. Koshelev, On kernel functions for the Hilbert space of polyanalytic functions in the disk. Dokl. Akad. Nauk. SSSR [Soviet Math. Dokl.] 232, 277–279 (1977)Google Scholar
  57. 57.
    G. Kuttyniok, Affine Density in Wavelet Analysis, Lecture Notes in Mathematics, vol. 1914 (Springer, Berlin, 2007)Google Scholar
  58. 58.
    Y. Lyubarskii, P.G. Nes, Gabor frames with rational density. Appl. Comp. Harm. Anal. 34, 488–494 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Y. Lyubarskii, Frames in the Bargmann space of entire functions, Entire and subharmonic functions, Adv. Soviet Math., vol. 11 (Amer. Math. Soc., Providence, RI, 1992), pp. 167–180Google Scholar
  60. 60.
    T. Mine, Y. Nomura, Landau levels on the hyperbolic plane in the presence of Aharonov–Bohm fields. J. Funct. Anal. 263, 1701–1743 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    M.L. Mehta, Random Matrices, 3rd edn. (Academic Press, New York, 1991)zbMATHGoogle Scholar
  62. 62.
    Z. Mouayn, Characterization of hyperbolic Landau states by coherent state transforms. J. Phys. A Math. Gen. 36, 8071–8076 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    N.I. Muskhelishvili, Some Basic Problems of Mathematical Elasticity Theory (in Russian) (Nauka, Moscow, 1968)Google Scholar
  64. 64.
    A.M. Perelomov, On the completeness of a system of coherent states. Theor. Math. Phys. 6, 156–164 (1971)MathSciNetCrossRefGoogle Scholar
  65. 65.
    A.M. Perelomov, Generalized Coherent States and Their Applications (Springer, New York, 1986). Texts Monographs Phys. 6, 156–164 (1971)MathSciNetGoogle Scholar
  66. 66.
    J. Ramanathan, T. Steger, Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2, 148–153 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    A.K. Ramazanov, On the structure of spaces of polyanalytic functions. (Russian) Mat. Zametki 72(5), 750–764 (2002); translation in Math. Notes 72(5–6), 692–704 (2002)Google Scholar
  68. 68.
    W. Roelcke, Der eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I. Math. Ann. 167, 292–337 (1966)MathSciNetCrossRefGoogle Scholar
  69. 69.
    A. Ron, Z. Shen, Weyl-Heisenberg frames and Riesz bases in \({L}^{2}({\mathbb{R}}^{d})\). Duke Math. J. 89, 237–282 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    K. Seip, Reproducing formulas and double orthogonality in Bargmann and Bergman spaces. SIAM J. Math. Anal. 22(3), 856–876 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    K. Seip, R. Wallstén, Density Theorems for sampling and interpolation in the Bargmann-Fock space II. J. Reine Angew. Math. 429, 107–113 (1992)MathSciNetzbMATHGoogle Scholar
  72. 72.
    K. Seip, Beurling type density theorems in the unit disc. Invent. Math. 113, 21–39 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    D. Slepian, Some comments on Fourier analysis, uncertainty and modelling. SIAM Rev. 25, 379–393 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, vol. 42 (Princeton University Press, New Jersey, 1993)Google Scholar
  75. 75.
    N.L. Vasilevski, On the structure of Bergman and poly-Bergman spaces. Integr. Equat. Oper. Theory 33, 471–488 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    N.L. Vasilevski, Poly-Fock Spaces, Differential operators and related topics, vol. I (Odessa, 1997), pp. 371–386, Oper. Theory Adv. Appl., vol. 117 (Birkhäuser, Basel, 2000)Google Scholar
  77. 77.
    N.L. Vasilevski, Commutative Algebras of Toeplitz Operators On the Bergman Space, Oper. Theory Adv. Appl., vol. 185 (Birkhäuser, Basel, 2008)Google Scholar
  78. 78.
    A. Wünsche, Displaced Fock states and their connection to quasiprobabilities. Quantum Opt. 3, 359–383 (1991)MathSciNetCrossRefGoogle Scholar
  79. 79.
    K. Zhu, Analysis on Fock Spaces. Graduate Texts in Mathematics, vol. 263 (Springer, New York, 2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Acoustic Research InstituteAustrian Academy of SciencesWienAustria
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  3. 3.NuHAG, Faculty of MathematicsUniversity of ViennaWienAustria

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