SMOOTH REFINABLE FUNCTIONS PROVIDE GOOD APPROXIMATION ORDERS

We apply the general theory of approximation orders of shift-invariant spaces of [C. de Boor, R. A. DeVore, and A. Ron, Trans Amer. Math. Se c., 341 (1994), pp. 787-806], [C. de Boor, R. A. DeVore, and A. Ron, J . Funct. Anal., 119 (1994), pp. 37-78], and [C. de Boor, R. A. DeVore, and A. Ron, ''Approximation orders of FSI spaces,'' Constr. Approx., 13 (1997), to appear] to the special case when the finitely many gener ators Phi subset of L-2(R-d) of the underlying space S satisfy an N-sc ale relation (i.e., they form a ''father-wavelet'' set). We show that the approximation orders provided by such finitely generated shift-inv ariant spaces are bounded from below by the smoothness class of each p si is an element of S (in particular, each phi is an element of Phi), as well as by the decay rate of it; Fourier transform. In fact, simila r results are valid for refinable shift-invariant spaces that are not finitely generated. Specifically, it is shown that under some mild tec hnical conditions on the scaling functions Phi, approximation order k is provided if either some psi, is an element of S lies in the Sobolev space W-2(k-1) or its Fourier transform <(psi)over cap>(w) decays nea r infinity like o(\w\(1-k)). No technical side conditions are required if the spatial dimension is d = i, and the functions in Phi are compa ctly supported. For the special case of a singleton Phi, our first cla ss of results (which are concerned with the condition phi is an elemen t of W-2(k-1)) improve previously known results of Meyer and Cavaretta , Dahmen, and Micchelli.