A geometric approach to the cascade approximation operator for wavelets
| dc.creator | Jorgensen, Palle E. T. | |
| dc.date | 1999-12-15 | |
| dc.date.accessioned | 2026-07-07T05:32:19Z | |
| dc.date.available | 2026-07-07T05:32:19Z | |
| dc.description | This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically the cascade algorithm in wavelet theory. Let $ H $ be a Hilbert space, and let $ π$ be a representation of $ L^\infty(T) $ on $ H $. Let $ R $ be a positive operator in $ L^\infty(T) $ such that $ R(1)=1 $, where $ 1 $ denotes the constant function $ 1 $. We study operators $ M $ on $ H $ (bounded, but non-contractive) such that $ π(f)M=Mπ(f(z^2)) $ and $ M^* π(f)M=π(R^* f) $, $ f \in L^\infty (T) $, where the $ * $ refers to Hilbert space adjoint. We give a complete orthogonal expansion of $ H $ which reduces $ π$ such that $ M $ acts as a shift on one part, and the residual part is $ H^{(\infty)}=\bigcap_n[M^n H] $, where $ [M^n H] $ is the closure of the range of $ M^n $. The shift part is present, we show, if and only if $ \ker(M^*) \neq \{0\} $. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation $ π$, we show that, for this wavelet operator $ M $, the components in the decomposition are unitarily, and canonically, equivalent to spaces $ L^2(E_n) \subset L^2(R) $, where $ E_n \subset R $, $ n=0,1,2,...,\infty $, are measurable subsets which form a tiling of $ R $; i.e., the union is $ R $ up to zero measure, and pairwise intersections of different $ E_n $'s have measure zero. We prove two results on the convergence of the cascade algorithm, and identify singular vectors for the starting point of the algorithm. | |
| dc.description | AMS-LaTeX; 47 pages, 3 tables, 2 figures comprising 3 EPS diagrams | |
| dc.identifier | https://arxiv.org/abs/math/9912132 | |
| dc.identifier | http://arxiv.org/abs/math/9912132 | |
| dc.identifier | Integral Equations Operator Theory 35 (1999), 125-171 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79620 | |
| dc.subject | Functional Analysis | |
| dc.subject | 46L60, 47D25, 42A16, 43A65 (Primary) 46L45, 42A65, 41A15 (Secondary) | |
| dc.title | A geometric approach to the cascade approximation operator for wavelets | |
| dc.type | text |