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Approximate approximations from scattered data. (English) Zbl 1119.41015
In this interesting paper, the authors extend the approximate multivariate quasi-interpolation on a uniform grid by dilated shifts of a smooth, rapidly decaying function [see e.g. V. Maz’ya and 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.

MSC:
41A30 Approximation by other special function classes
65D15 Algorithms for approximation of functions
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A25 Rate of convergence, degree of approximation
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