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On Generalized Walsh Bases

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Abstract

This paper continues the study of orthonormal bases (ONB) of \(L^{2}[0,1]\) introduced in Dutkay et al. (J. Math. Anal. Appl. 409(2):1128–1139, 2014) by means of Cuntz algebra \(\mathcal{O}_{N}\) representations on \(L^{2}[0,1]\). For \(N=2\), one obtains the classic Walsh system. We show that the ONB property holds precisely because the \(\mathcal{O}_{N}\) representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal processing we find a fast generalized transform and compare this generalized transform with the classic one with respect to compression and sparse signal recovery.

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Acknowledgements

This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL, 32816-1364, USA

    Dorin Ervin Dutkay

  2. Department of Mathematical Sciences, University of South Dakota, 414 E. Clark Street, Vermillion, SD, 57069, USA

    Gabriel Picioroaga

  3. Division of Applied Mathematics, The School of Education, Culture and Communication (UKK), Mälardalen University, Box 883, 721 23, Västerås, Sweden

    Sergei Silvestrov

Authors
  1. Dorin Ervin Dutkay
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  2. Gabriel Picioroaga
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  3. Sergei Silvestrov
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Correspondence to Gabriel Picioroaga.

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Dutkay, D.E., Picioroaga, G. & Silvestrov, S. On Generalized Walsh Bases. Acta Appl Math 163, 73–90 (2019). https://doi.org/10.1007/s10440-018-0214-x

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