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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 kernelon the $(1/n)$-spaced infinite grid. The archetypal example here isapproximation using shifts of a Gaussian kernel. We present new resultsconcerning interpolation of functions of exponential type, in particular,polynomials on the integer grid as a step en route to solve the generalinterpolation problem. For the Gaussian kernel we introduce a new class ofpolynomials, closely related to the probabilistic Hermite polynomials and showthat evaluations of the polynomials at the integer points provide thecoefficients of the interpolants. Taking cue from the classical Newtonpolynomial interpolation, we derive a closed formula for the Gaussianinterpolant of a continuous function on a uniform grid in the unit interval.

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