The shape parameter in the shifted surface spline III

This is a continuation of our study about shape parameter, based on an approach very different from that of Luh [1] and [2]. Here we adopt an error bound of convergence order O(dω1/d) as d→0d→0, where 0<ω<10<ω<1 is a constant and d   denotes essentially the fill-distance. The constant ωω is much smaller than the one appears in Luh [1] and [2] where the error bound is O(ω1/d)O(ω1/d) only. Moreover, the constant ωω here only mildly depends on the dimension n. It means that for high-dimensional problems the criteria of choosing the shape parameter presented in this paper are much better than those of Luh [1] and [2]. The drawback is that the distribution of data points must be slightly controlled.