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Approximation orders of shift-invariant subspaces of \(W_{2}^{s}(\mathbb R^d)\). (English) Zbl 1092.41010

The authors discuss approximation orders of shift-invariant spaces in Sobolev spaces. First, they characterize in Theorem 21 the approximation orders provided by finitely generated shift-invariant (FSI) spaces by means of the Fourier transforms of the generators, extending a result of de C. Boor, R. A. de Vore and A. Ron [Trans. Am. Math. Soc. 341, No. 2, 787–806 (1994; Zbl 0790.41012)] from the \(L^2\) space to Sobolev spaces. Then they prove the existence of a superfunction and discuss its behavior (good or bad superfunction). Finally, approximation orders, Strang-Fix type conditions, and polynomial reproductions are investigated for FSI generators associated with matrix refinement equations.

MSC:

41A25 Rate of convergence, degree of approximation
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Citations:

Zbl 0790.41012
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References:

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