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A characterization of the approximation order of multivariate spline spaces. (English) Zbl 0722.41016

Summary: We analyze the approximation order associated with a directed set of spaces, \(\{S_ h\}_{h>0}\), each of which is spanned by the \(h{\mathbb{Z}}^ s\)-translates of one compactly supported function \(\phi_ h: {\mathbb{R}}^ s\to {\mathbb{C}}\). Under a regularity condition on the sequence \(\{\phi_ h\}_ h\), we show that the optimal approximation order (in the \(\infty\)-norm) is always realized by quasi-interpolants, hence in a linear way. These quasi-interpolants provide the best approximation rates from \(\{S_ h\}_ h\) to an exponential space of good approximation order at the origin. As for the case when \(S_ k\) is obtained by scaling \(S_ 1\), under the assumption \((*)\quad \sum_{\alpha \in {\mathbb{Z}}^ s}\phi_ 1(\cdot -\alpha)\not\equiv 0,\) the results here provide an unconditional characterization of the best approximation order in terms of the polynomials in \(S_ 1\). The necessity of (*) in this characterization is demonstrated by a counterexample.

MSC:

41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
41A40 Saturation in approximation theory
41A63 Multidimensional problems
65D15 Algorithms for approximation of functions
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