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 [1,2]. Here we adopt an error bound of convergence order O ( d ω 1 / d ) as d → 0 , where 0 < ω < 1 is a constant and d denotes essentially the fill-distance. The constant ω is much smaller than the one appears in Luh [1,2] where the error bound is O ( ω 1 / d ) only. Moreover, the constant ω 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 [1,2]. The drawback is that the distribution of data points must be slightly controlled.