Realizations of infinite products, Ruelle operators and wavelet filters

TL;DR

This paper develops explicit realization formulas for infinite products of rational matrix-functions, connecting systems theory, wavelet filters, and Ruelle operators, with applications to wavelet generator Fourier transforms and transfer operator representations.

Contribution

It introduces a novel realization approach for infinite products of rational functions, linking system realizations to wavelet theory and transfer operators in an infinite-dimensional setting.

Findings

Infinite product realizations are functions in an infinite-dimensional complex domain.

Realizations of infinite products are represented as infinite-dimensional Toeplitz operators.

Constructed infinite products realize Fourier transforms of wavelet generators and matrix representations of Ruelle operators.

Abstract

Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations of these resulting infinite products have the following four features: 1) Our infinite product realizations are functions 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 a symbol that is a reflection of the initial realization for $M$. 3) Starting with a subclass of rational matrix functions, including scalar-valued 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 produce 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.