WIAS Preprint No. 1058, (2005)

Approximate approximations from scattered data


  • Lanzara, Flavia
  • Maz'ya, Vladimir
  • Schmidt, Gunther

2010 Mathematics Subject Classification

  • 41A30 65D15 41A63 41A25


  • scattered data quasi-interpolation, cubature of integral operators, multivariate approximation, error estimates




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 on a uniform grid 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 the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.

Appeared in

  • J. Approx. Theory, 145 (2007), pp. 141--170

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