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Minimality of the data in wavelet filters | | Palle E.T. Jorgensen
; | | Date: |
14 Apr 2000 | | Journal: | Adv. Math. 159 (2001), 143--228 DOI: 10.1006/aima.2000.1958 | | Subject: | Functional Analysis; Mathematical Physics MSC-class: 46L60, 47D25, 42A16, 43A65 (Primary); 33C45, 42C10, 94A12, 46L45, 42A65, 41A15 (Secondary) | math.FA math-ph math.MP | | Abstract: | Orthogonal wavelets, or wavelet frames, for L^2(R) are associated with quadrature mirror filters (QMF). The latter constitute a set of complex numbers which relate the dyadic scaling of functions on R to the Z-translates, and which satisfy the QMF-axioms. In this paper, we show that generically, the data in the QMF-systems of wavelets is minimal, in the sense that it cannot be nontrivially reduced. The minimality property is given a geometric formulation in the Hilbert space l^2(Z), and it is then shown that minimality corresponds to irreducibility of a wavelet representation of the algebra O_2; and so our result is that this family of representations of O_2 on the Hilbert space l^2(Z) is irreducible for a generic set of values of the parameters which label the wavelet representations. | | Source: | arXiv, math.FA/0004098 | | Services: | Forum | Review | PDF | Favorites | |
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