48 research outputs found
Realizations of infinite products, Ruelle operators and wavelet filters
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 , we show that a resulting infinite product
realization obtained from takes the form of an (infinite-dimensional)
Toeplitz operator with a symbol that is a reflection of the initial realization
for . 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 wavelets. 4) We use both the
realizations for 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
Realization of Tensor Product and of Tensor Factorization of Rational Functions
We study the state space realization of a tensor product of a pair of rational functions. At the expense of “inflating” the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, we introduce an explicit formula for a tensor factorization of this function