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Smooth Refinable Functions Provide Good Approximation Orders

By Amos Ron


We apply the general theory of approximation orders of shift-invariant spaces of [BDR1-3] to the special case when the finitely many generators Φ ⊂ L2(IR d) of the underlying space S satisfy an N-scale relation (i.e., they form a “father wavelet ” set). We show that the approximation orders provided by such finitely generated shift-invariant spaces are bounded from below by the smoothness class of each ψ ∈ S (in particular, each φ ∈ Φ), as well as by the decay rate of its Fourier transform. In fact, similar results are valid for refinable shift-invariant spaces that are not finitely generated. Specifically, it is shown that, under some mild technical conditions on the scaling functions Φ, approximation order k is provided if either some ψ ∈ S lies in the Sobolev space W k−1 2, or its Fourier transform ψ(w) decays near ∞ like o(|w | 1−k). No technical side-conditions are required if the spatial dimension is d = 1, and the functions in Φ are compactly supported. For the special case of a singleton Φ, our first class of results (that are concerned with the condition φ ∈ W k−1 2) improve previously known results of Yves Meyer and o

Topics: Key Words, Wavelets, scaling functions, father wavelet, approximation orders, principal
Year: 1995
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