Local accuracy for radial basis function interpolation on finite uniform grids

JOURNAL OF APPROXIMATION THEORY

卷:99

Article

Bejancu, A

关键词: radial basis function interpolation;    local error estimates;    finite uniform grids;   

DOI  :  10.1006/jath.1999.3332

来源: Elsevier

【 摘 要 】

We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) phi(r) = r(gamma) for gamma > 0, gamma is not an element of 2N or phi(r) = r(gamma) ln r for gamma is an element of 2N(+). For each positive integer N, let h = N-1 and let {x(i):i = 1, 2, ..., (N + 1)(d)} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1](d). Given f: [0, 1](d) --> R, let s(h) be its unique RBF interpolant at the grid vertices: s(h)(x(i)) = f(x(i)), i = 1, 2, ..., (N + 1)(d). For h --> 0, we show that the uniform norm of the error f-s(h) on a compact subset K of the interior of [0, 1](d) enjoys the same rate of convergence to zero as the error of RDF interpolation on the infinite uniform grid hZ(d), provided that f is a data function whose partial derivatives in the interior of [0, 1](d) up to a certain order can be extended to Lipschitz functions on [0, 1](d). (C) 1999 Academic Press.

【 授权许可】

Free