Realizations of Infinite Products, Ruelle Operators and Wavelet Filters

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 $$M$$M, we show that a resulting infinite product realization obtained from $$M$$M takes the form of an (infinite-dimensional) Toeplitz operator with the symbol that is a reflection of the initial realization for $$M$$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 $$\mathbf L_2(\mathbb R)$$L2(R) wavelets. (4) We use both the realizations for $$M$$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.