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Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

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Excursions in Harmonic Analysis, Volume 2

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

Abstract

In this chapter we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix-valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example, Krein spaces, in the study of stability questions. We focus on the nonrational case and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces and the Cuntz relations.

MSC classes:65T60, 46C20, 93B28

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Notes

  1. 1.

    Classically, in the engineering literature, the functions are analytic, or more generally meromorphic, outside the closed unit disk. The map z↦1 ∕ zrelates the two settings.

  2. 2.

    For rational functions, the term para-unitaryis also used in the engineering literature.

  3. 3.

    This correspondence: polynomial filterto compactly supported waveleteven works if d > 1.

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Acknowledgments

D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The work was done in part while the second named author visited Department of Mathematics, Ben Gurion University of the Negev, supported by a BGU distinguished visiting scientist program. Support and hospitality are much appreciated. We acknowledge discussions with colleagues there and in the USA, Dorin Dutkay, Myung–Sin Song, and Erin Pearse.

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Authors and Affiliations

  1. Department of Mathematics, Ben Gurion University of the Negev, 653, Be’er Sheva, 84105, Israel

    Daniel Alpay

  2. Department of Mathematics, The University of Iowa City, 14 MLH, Iowa City, IA, 52242-1419, USA

    Palle Jorgensen

  3. Department of Electrical Engineering, Ben Gurion University of the Negev, 653, Be’er Sheva, 84105, Israel

    Izchak Lewkowicz

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  1. Daniel Alpay
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  2. Palle Jorgensen
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Correspondence to Palle Jorgensen .

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Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

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Alpay, D., Jorgensen, P., Lewkowicz, I. (2013). Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8379-5_5

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