Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation

AbstractA class of multivariate scattered data interpolation methods which includes the so-called multiquadrics is considered. Pointwise error bounds are given in terms of several parameters including a parameter d which, roughly speaking, measures the spacing of the points at which interpolation occurs. In the multiquadric case these estimates are O(λ1d) as d → 0, where λ is a constant which satisfies 0 < λ < 1. An essential ingredient in this development which may be of independent interest is a bound on the size of a polynomial over a cube in Rn in terms of its values on a discrete subset which is scattered in a sufficiently uniform manner