Realizations of infinite products, Ruelle operators and wavelet filters. (English) Zbl 1326.42040
Authors’ 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.
Reviewer: Brett Wick (St. Louis)
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 65T60 | Numerical methods for wavelets |
| 47A48 | Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. |
| 40A20 | Convergence and divergence of infinite products |
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