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- W2008311911 abstract "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. Soc., 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 generators $Phisubset L_2(Rd)$ 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 $psiin S$ (in particular, each $phiinPhi$), 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 $Phi$, approximation order k is provided if either some $psiin S$ lies in the Sobolev space $W_2^{k-1}$ or its Fourier transform $hatpsi(w)$ decays near $infty$ like $o(|w|^{1-k})$. No technical side conditions are required if the spatial dimension is $d=1$, and the functions in $Phi$ are compactly supported. For the special case of a singleton $Phi$, our first class of results (which are concerned with the condition $phiin W^{k-1}_2$) improve previously known results of Meyer and Cavaretta, Dahmen, and Micchelli." @default.
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- W2008311911 date "1997-05-01" @default.
- W2008311911 modified "2023-09-26" @default.
- W2008311911 title "Smooth Refinable Functions Provide Good Approximation Orders" @default.
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- W2008311911 doi "https://doi.org/10.1137/s0036141095282486" @default.
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