Approximate approximations from scattered data
In this interesting paper, the authors extend the approximate multivariate quasi-inter\-po\-lation on a uniform grid by dilated shifts of a smooth, rapidly decaying function [see e.g. \textit{V. Maz'ya} and \textit{G. Schmidt}, IMA J. Numer. Anal. 16, 13--29 (1996; Zbl 0838.65005)] to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions, the authors present a constructive method for an approximate partition of unity and apply this method to the cubature of an integral operator with radial kernel. Numerical tests are given for scattered data quasi-interpolation of univariate functions and for approximate partitions of unity with Gaussians in the univariate/bivariate case.
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