On the High-Level Error Bound for Multiquadric and Inverse Multiquadric Interpolations

It’s well-known that there is a so-called high-level error bound for multiquadric and inverse multiquadric interpolations, which was put forward by Madych and Nelson in 1992. It’s of the form |f(x)− s(x)| ≤ λ 1 d ‖f‖h where 0 < λ < 1 is a constant, d is the fill distance which roughly speaking measures the spacing of the data points, s(x) is the interpolating function of f(x), and h denotes the multiquadric or inverse multiquadric. The error bound converges very fast as d → 0. The constant λ is very sensitive. A slight change of it will result in a huge change of the error bound. Unfortunately λ can not be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer the question.