A New Error Bound for Shifted Surface Spline Interpolation
Shifted surface spline is a frequently used radial function for scattered data interpolation. The most frequently used error bounds for this radial function are the one raised by Wu and Schaback in  and the one raised byMadych and Nelson in . Both are O(dl) as d → 0, where l is a positive integer and d is the well-known fill-distance which roughly speaking measures the spacing of the data points. Then RBF people found that there should be an error bound of the form O(ω 1 d ) because shifted surface spline is smooth and every smooth function shares this property. This only problem was that the value of the cucial constant ω was unknown. Recently Luh raised an exponential-type error bound with convergence rate O(ω 1 d ) as d → 0 where 0 < ω < 1 is a fixed constant which can be accurately computed . Although the exponential-type error bound converges much faster than the algebraic-type error bound, the constant ω is intensely influenced by the dimension n in the sense ω → 1 rapidly as n → ∞. Here the variable x of both the interpolated and interpolating functions lies in Rn. In this paper we present an error bound which is O(√dω′1d ) where 0 < ω′ < 1 is a fixed constant for any fixed n, and is only mildly influenced by n. In other words, ω′ → 1 very slowly as n → ∞, and ω′
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