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Construction techniques for highly accurate quasi-interpolation operators. (English) Zbl 0904.41012
The authors consider univariate quasi-interpolants of the form \[ f_h(x)= \sum^{+\infty}_{-\infty} f(hj)\varphi_h(x/h- j), \] for \(x\in\mathbb{R}\) and \(h>0\), where \(\varphi_h\) is in turn a linear combination of translates \(\psi(x- jh)\) of a function \(\psi\) in \(C^\ell(\mathbb{R})\). Thus the sampling distance of the data \(f(jh)\) is actually different from the shifts used for the function \(\psi\). It is shown that it is possible to find linear combinations \(\varphi_h\) such that the order of convergence of the quasi-interpolants is only limited by the smoothness of the function \(\psi\). As the authors’ technique involves discrete convolutions with \(B\)-splines, it can be generalized to the multivariate setting by using discrete convolutions with tensor-products of odd-degree \(B\)-splines.
Reviewer: E.Quak (Oslo)

MSC:
41A30 Approximation by other special function classes
41A05 Interpolation in approximation theory
41A25 Rate of convergence, degree of approximation
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