Hermite function interpolation on a finite uniform grid: Defeating the Runge phenomenon and replacing radial basis functions

We show that Hermite functions are successful for interpolation on a finite interval, even on a uniform grid where polynomial interpolation fails. The Runge Phenomenon is not completely abolished, but is greatly diminished. Finite interval Hermite interpolation is ill-conditioned and therefore limited to a maximum of about 250 interpolation points on a uniform univariate grid, but it is still far superior to the approach using Gaussian radial basis functions (RBFs). The Hermite functions are a complete spectral basis for the infinite interval; the motivation for employing them here derives from a careful study that showed, for small RBF shape parameter alpha and small number of points No that Gaussian RBF cardinal functions can be accurately approximated by the product of a Gaussian with the usual polynomial cardinal function. Direct comparisons show that when N and alpha are not both small, Hermite interpolation is greatly superior in accuracy, condition number and efficiency to the RBF methods that inspired it. (C) 2013 Elsevier Ltd. All rights reserved.