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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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