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 α and small number of points N , 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 α are not both small, Hermite interpolation is greatly superior in accuracy, condition number and efficiency to the RBF methods that inspired it.