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

Abstract

<p style='text-indent:20px;'>In this paper, we present a new approach to solving the problem of interpolating a continuous function at <inline-formula><tex-math id="M1">\begin{document}$ (n+1) $\end{document}</tex-math></inline-formula> equally-spaced points in the interval <inline-formula><tex-math id="M2">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula>, using shifts of a kernel on the <inline-formula><tex-math id="M3">\begin{document}$ (1/n) $\end{document}</tex-math></inline-formula>-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. Finally we give a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval (assuming knowledge of the discrete moments of the Gaussian).