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The \(\alpha \)-modulation transform: admissibility, coorbit theory and frames of compactly supported functions

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Abstract

The \(\alpha \)-modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl–Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in \(\alpha \)-modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results.

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Acknowledgements

This work was funded by the Austrian Science Fund (FWF) DACH-Project BIOTOP(‘Adaptive Wavelet and Frame techniques for acoustic BEM’; I-1018-N25), by the FWF START-Project FLAME (‘Frames and Linear Operators for Acoustical Modeling and Parameter Estimation’; Y 551-N13) and the DFG Project Number DA 360/19-1. We would like to thank all the Project members for valuable discussions and comments.

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Authors and Affiliations

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

    M. Speckbacher & P. Balazs

  2. Inst. für Mathematik und Informatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    D. Bayer

  3. Philipps-Universität Marburg, FB12 Mathematik und Informatik, Hans-Meerwein Straße, Lahneberge, 35032, Marburg, Germany

    S. Dahlke

Authors
  1. M. Speckbacher
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  2. D. Bayer
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  3. S. Dahlke
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  4. P. Balazs
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Correspondence to M. Speckbacher.

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Communicated by A. Constantin.

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Speckbacher, M., Bayer, D., Dahlke, S. et al. The \(\alpha \)-modulation transform: admissibility, coorbit theory and frames of compactly supported functions. Monatsh Math 184, 133–169 (2017). https://doi.org/10.1007/s00605-017-1085-3

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  • DOI: https://doi.org/10.1007/s00605-017-1085-3

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