## Abstract

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.

Original language | English |
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Pages (from-to) | 1034-1052 |

Number of pages | 19 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 21 |

Issue number | 5 |

DOIs | |

State | Published - 7 Oct 2015 |

## Keywords

- Filter banks
- Infinite products
- State space realization
- Wavelet filters

## ASJC Scopus subject areas

- Analysis
- Mathematics (all)
- Applied Mathematics