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

Daniel Alpay; Palle Jorgensen; Izchak Lewkowicz

doi:https://doi.org/10.1007/978-0-8176-8379-5_5

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

Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Ben-Gurion University of the Negev

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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.

Original language	English
Title of host publication	Excursions in Harmonic Analysis
Subtitle of host publication	The February Fourier Talks at the Norbert Wiener Center
Editors	Travis D. Andrews, Radu Balan, John J. Benedetto, Wojciech Czaja, Kasso A. Okoudjou
Publisher	Birkhauser Boston
Pages	69-111
Number of pages	43
Volume	2
ISBN (Electronic)	9780817683795
ISBN (Print)	9780817683788
DOIs	https://doi.org/10.1007/978-0-8176-8379-5_5
State	Published - 2013

Keywords

Cuntz relations
Pontryagin spaces
Schur analysis
Wavelet filters

All Science Journal Classification (ASJC) codes

Applied Mathematics

Access to Document

https://doi.org/10.1007/978-0-8176-8379-5_5

Cite this

Alpay, D., Jorgensen, P., & Lewkowicz, I. (2013). Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces. In T. D. Andrews, R. Balan, J. J. Benedetto, W. Czaja, & K. A. Okoudjou (Eds.), Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center (Vol. 2, pp. 69-111). Birkhauser Boston. https://doi.org/10.1007/978-0-8176-8379-5_5

Alpay, Daniel ; Jorgensen, Palle ; Lewkowicz, Izchak. / Extending Wavelet Filters : Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces. Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. editor / Travis D. Andrews ; Radu Balan ; John J. Benedetto ; Wojciech Czaja ; Kasso A. Okoudjou. Vol. 2 Birkhauser Boston, 2013. pp. 69-111

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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.",

keywords = "Cuntz relations, Pontryagin spaces, Schur analysis, Wavelet filters",

author = "Daniel Alpay and Palle Jorgensen and Izchak Lewkowicz",

note = "Funding Information: 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. Funding Information: 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. Publisher Copyright: {\textcopyright} Springer Science+Business Media New York 2013.",

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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 TD Andrews, R Balan, JJ Benedetto, W Czaja & KA Okoudjou (eds), Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. vol. 2, Birkhauser Boston, pp. 69-111. https://doi.org/10.1007/978-0-8176-8379-5_5

Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces. / Alpay, Daniel; Jorgensen, Palle; Lewkowicz, Izchak.
Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. ed. / Travis D. Andrews; Radu Balan; John J. Benedetto; Wojciech Czaja; Kasso A. Okoudjou. Vol. 2 Birkhauser Boston, 2013. p. 69-111.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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T1 - Extending Wavelet Filters

T2 - Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

AU - Alpay, Daniel

AU - Jorgensen, Palle

AU - Lewkowicz, Izchak

N1 - Funding Information: 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. Funding Information: 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. Publisher Copyright: © Springer Science+Business Media New York 2013.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Cuntz relations

KW - Pontryagin spaces

KW - Schur analysis

KW - Wavelet filters

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U2 - https://doi.org/10.1007/978-0-8176-8379-5_5

DO - https://doi.org/10.1007/978-0-8176-8379-5_5

M3 - Chapter

SN - 9780817683788

VL - 2

SP - 69

EP - 111

BT - Excursions in Harmonic Analysis

A2 - Andrews, Travis D.

A2 - Balan, Radu

A2 - Benedetto, John J.

A2 - Czaja, Wojciech

A2 - Okoudjou, Kasso A.

PB - Birkhauser Boston

ER -

Alpay D, Jorgensen P, Lewkowicz I. Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces. In Andrews TD, Balan R, Benedetto JJ, Czaja W, Okoudjou KA, editors, Excursions in Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Center. Vol. 2. Birkhauser Boston. 2013. p. 69-111 doi: https://doi.org/10.1007/978-0-8176-8379-5_5

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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