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Function Spaces of Polyanalytic Functions

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Harmonic and Complex Analysis and its Applications

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Abstract

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.

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Acknowledgments

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.

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Authors and Affiliations

  1. Acoustic Research Institute, Austrian Academy of Sciences, Wohllebengasse 12–14, Wien, A-1040, Austria

    Luis Daniel Abreu

  2. CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal

    Luis Daniel Abreu

  3. NuHAG, Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Wien, Austria

    Hans G. Feichtinger

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  1. Luis Daniel Abreu
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Correspondence to Luis Daniel Abreu .

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

    Alexander Vasil'ev

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Abreu, L.D., Feichtinger, H.G. (2014). Function Spaces of Polyanalytic Functions. 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_1

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