  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.
