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Hyperbolic Wavelet Frames and Multiresolution in the Weighted Bergman Spaces

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Landscapes of Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In this paper, we construct so-called hyperbolic wavelet frames in weighted Bergman spaces and a multiresolution analysis (MRA) generated by them. The construction is based on a new example of sampling set for the weighted Bergman space, which is related to the Blaschke group operation. The introduced MRA is an analog of the MRA generated by the affine wavelets in the space of the square integrable functions on the real line, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the weighted Bergman space. The projection to the resolution levels is an interpolation operator. This projection operator gives opportunity of practical realization of the hyperbolic wavelet representation of a function belonging to the weighted Bergman space, if we can measure the values of the function on a given set of points inside the unit disc. Convergence properties of the hyperbolic wavelet representation are studied.

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Acknowledgements

This research was supported by the grant EFOP-3.6.1.-16-2016-00004 Comprehensive Development for Implementing Smart Specialization Strategies at the University of Pécs.

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Authors and Affiliations

  1. University of Pécs, Ifjúság útja 6, Pécs, 7634, Hungary

    Margit Pap

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  1. Margit Pap
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Correspondence to Margit Pap .

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Editors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Paolo Boggiatto

  2. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Elena Cordero

  3. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Maurice de Gosson

  4. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Fabio Nicola

  6. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Anita Tabacco

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Pap, M. (2019). Hyperbolic Wavelet Frames and Multiresolution in the Weighted Bergman Spaces. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05210-2_9

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