Approximate approximations from scattered data

Authors:
F. Lanzara;V. Maz'ya;G. Schmidt
Affiliations:
Dipartimento di Matematica, Università “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy;Department of Mathematics, University of Linköping, 581 83 Linköping, Sweden and Department of Mathematics, Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA and Departme ...;Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
Venue:
Journal of Approximation Theory
Year:
2007

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Abstract

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function 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 we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.