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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(Ψ)={ψ j,k =2 jd/2 ψ(2 j y-k):ψΨ,jZ,kZ d }L 2 (R d ). A wavelet is said to be MRA-based if there exists a multiresolution analysis such that ΨV 1 . If X(Ψ) 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) i=0 l m+l isin 2i (x/2)cos 2(l-i) (x/2). Fast implementation algorithms are also provided.

MSC:
42C40Wavelets and other special systems
42C15General harmonic expansions, frames