2014/06/20 by Daniel Alpay, Alpay, Daniel, Palle E. T. Jørgensen +3
Computer Science · Engineering · Mathematics · #40A20 #42C40 #47A48 #67T60 #Advanced Numerical Analysis Techniques #Complex Variables (math.CV) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1406.5338
openalex publication_date 2014/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2(\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.