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Fourier analysis of the approximation power of principal shift-invariant spaces. (English) Zbl 0801.41027
Summary: The approximation order provided by a directed set $$\{s_ h\}_{h>0}$$ of spaces, each spanned by the $$h\mathbb{Z}^ d$$-translates of one function, is analyzed. The “near-optimal” approximants of [R2] from each $$s_ h$$ to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.
The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale $$\{s_ h\}$$ is obtained from $$s_ 1$$ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 42A10 Trigonometric approximation 41A15 Spline approximation
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