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 L2​(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.Comment: corrected versio