Approximation of exponential-type functions on a uniform grid by shifts of a basis function

TL;DR

This paper develops a new interpolation method for exponential-type functions using shifts of a basis function, particularly Gaussian kernels, on a uniform grid, introducing novel polynomials related to Hermite polynomials.

Contribution

It introduces a new class of polynomials for Gaussian kernel interpolation and provides a closed-form formula for the interpolant on a uniform grid.

Findings

New polynomial class related to Hermite polynomials

Closed-form Gaussian interpolant formula

Effective interpolation on uniform grids

Abstract

In this paper, we study the problem of interpolating a continuous function at $(n+1)$ equally-spaced points in the interval $[0,1]$, using shifts of a kernel on the $(1/n)$-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Taking cue from the classical Newton polynomial interpolation, we derive a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval.