A New Error Bound for Shifted Surface Spline Interpolation

Abstract

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 [17] and the one raised byMadych and Nelson in [14]. 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 [11]. 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′