Unitary representations of wavelet groups and encoding of iterated function systems in solenoids (original) (raw)
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We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [DJ06a], and we present several equivalent formulations of the problem. 2000 Mathematics Subject Classification. 42C40 ,28D05,47A67,28A80 .
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2000
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Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C∗-algebra ON . A main tool in our analysis is the infinite-dimensional group of all maps T → U (N) (where U (N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.
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2000
In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C^*-algebra O_N. A main tool in our analysis is the infinite-dimensional group of all maps T -> U(N) (where U(N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.
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2013
We here use notions from the theory linear shift-invariant dynamical systems to provide an explicit characterization, both practical and computable, of all rational wavelet filters. For a given N , (N ≥ 2) the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree N 1 2 (N − 1) + m. Moreover, beyond the parameters N and m, one confine the spectrum of the filters to lie in an open disk of radius ρ (stable filters mean ρ ∈ [0, 1] and for FIR take ρ = 0). Then all filters can be described by a convex set of parameters [0, π) × [0, 2π) 2(N−1) × [0, ρ) m. Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parametrization is exploited for a step-by-step construction (where in each, the index m is increased by one) of state space model of wavelet filters.
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