An Extension of the Exponential-type Error Bounds for Multiquadric and Gaussian Interpolations

In the 1990’s exponential-type error bounds appeared in the theory of radial basis functions. For multiquadric interpolation it is O(λ 1 d ) as d → 0, where λ is a constant satisfying 0 < λ < 1. For Gaussian interpolation it is O(C d) c′ d as d → 0 where C ′ and c are constants. In both cases the parameter d, called fill distance, measures the spacing of the points where interpolation occurs. This kind of error bounds is very powerful. However it only measures the difference between the approximant and approximand. Mathematicians and engineers often need to know the matching of the derivatives when dealing with partial differential equations. In this paper we extend this kind of error bounds to a form which measures the difference between the derivatives of the approximant and approximand. keywords: radial basis function, error bound, multiquadric, Gaussian, derivative. AMS subject classification:41A05,41A25,41A30,41A63,65D10.