  In this paper we present a new multilevel quasi-interpolation algorithm for
smooth periodic functions using scaled Gaussians as basis functions. Recent
research in this area has focussed upon implementations using basis function
with finite smoothness. In this paper we deliver a first error estimates for
the multilevel algorithm using analytic basis functions. The estimate has two
parts, one involving the convergence of a low degree polynomial truncation term
and one involving the control of the remainder of the truncation as the
algorithm proceeds. Thus, numerically one observes a convergent scheme.
Numerical results suggest that the scheme converges much faster than the theory
shows.
