Smooth Reenable Functions Provide Good Approximation Orders Smooth Reenable Functions Provide Good Approximation Orders

We apply the general theory of approximation orders of shift-invariant spaces of BDR1-3] to the special case when the nitely many generators L 2 (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 nitely generated shift-invariant spaces are bounded from below by the smoothness class of each 2 S (in particular, each 2), as well as by the decay rate of its Fourier transform. In fact, similar results are valid for reenable shift-invariant spaces that are not nitely generated. Speciically, it is shown that, under some mild technical conditions on the scaling functions , approximation order k is provided if either some 2 S lies in the Sobolev space W k?1 2 , or its Fourier transform b (w) decays near 1 like o(jwj 1?k). No technical side-conditions are required if the spatial dimension is d = 1, and the functions in are compactly supported. For the special case of a singleton , our rst class of results (that are concerned with the condition 2 W k?1 2) improve previously known results of Yves Meyer and of Cavaretta-Dahmen-Micchelli.