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Framelets: MRA-based constructions of wavelet frames. (English) Zbl 1035.42031

A wavelet system is a collection of the form \(X(\Psi) = \{ \psi_{j,k} = 2^{jd/2} \psi(2^jy-k): \psi \in \Psi, j \in Z, k \in Z^d \} \subset L^2(R^d)\). A wavelet is said to be MRA-based if there exists a multiresolution analysis such that \(\Psi \subset V_1\). If \(X(\Psi)\) is a frame, we call its elements {framelets}. The authors present the unitary extension principle and the oblique extension principle to facilitate constructions of MRA-based tight wavelet frames. Approximation orders and vanishing moments of MRA-based wavelet systems are studied in order to construct framelets with higher approximation orders. The scaling functions of these framelets are pseudo-splines, i.e., square roots of \(\cos^{2m}(x/2) \sum_{i=0}^l \binom{m+l}{i} \sin^{2i}(x/2) \cos^{2(l-i)}(x/2)\). Fast implementation algorithms are also provided.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
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