An Improved Error Bound for Gaussian Interpolation

It’s well known that there is a so-called exponential-type error bound for Gaussian interpolation which is the most powerful error bound hitherto. It’s of the form |f(x) − s(x)| ≤ c1(c2d) c3 d ‖f‖h where f and s are the interpolated and interpolating functions respectively, c1, c2, c3 are positive constants, d is the fill-distance which roughly speaking measures the spacing of the data points, and ‖f‖h is the h-norm of f where h is the Gaussian function. The error bound is suitable for x ∈ R, n ≥ 1, and gets small rapidly as d → 0. The drawback is that the crucial constants c2 and c3 get worse rapidly as n increases in the sense c2 → ∞ and c3 → 0 as n → ∞. In this paper we raise an error bound of the form |f(x)− s(x)| ≤ c1(c2d) c′ 3 d √ d‖f‖h, where c2 and c3 are independent of the dimension n. Moreover, c2 << c2, c3 << c ′ 3, and c ′ 1 is only slightly different from c1. What’s important is that all constants c1, c ′ 2 and c ′ 3 can be computed without slight difficulty. AMS classification:41A05,41A25,41A30,41A63,65D10. keywords:radial basis function, interpolation, error bound, Gaussian.