Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids |
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Authors: | Aurelian Bejancu Jr. |
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Affiliation: | Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, Englandf1 |
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Abstract: | We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=rγ for γ>0, γ2 or φ(r)=rγ ln r for γ2+. For each positive integer N, let h=N−1 and let {xi: 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→, let sh be its unique RBF interpolant at the grid vertices: sh(xi)=f(xi), i=1, 2, …, (N+1)d. For h→0, we show that the uniform norm of the error f−sh on a compact subset K of the interior of [0, 1]d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid hd, 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. |
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Keywords: | radial basis function interpolation local error estimates finite uniform grids |
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