Abstract
Using the system theory notion of state-space realization of matrix-valued rational functions, we describe the Ruelle operator associated with wavelet filters. The resulting realization of infinite products of rational functions have the following four features: (1) It is defined in an infinite-dimensional complex domain. (2) Starting with a realization of a single rational matrix-function \(M\), we show that a resulting infinite product realization obtained from \(M\) takes the form of an (infinite-dimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for \(M\). (3) Starting with a subclass of rational matrix functions, including scalar-valued ones corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of \(\mathbf L_2(\mathbb R)\) wavelets. (4) We use both the realizations for \(M\) and the corresponding infinite product to obtain a matrix representation of the Ruelle-transfer operators used in wavelet theory. By “matrix representation” we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration.
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Acknowledgments
D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The authors thank the US-Israel Binational Science Foundation (BSF) Grant number 2010117. It is a pleasure to thank the referees for a very careful reading of the manuscript.
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Communicated by Karlheinz Gröchenig.
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Alpay, D., Jorgensen, P. & Lewkowicz, I. Realizations of Infinite Products, Ruelle Operators and Wavelet Filters. J Fourier Anal Appl 21, 1034–1052 (2015). https://doi.org/10.1007/s00041-015-9396-z
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DOI: https://doi.org/10.1007/s00041-015-9396-z