On approximations with Gaussian and related functions

The Gaussians belong to a class of functions which are smooth, decay rapidly and have the property that the application of the integral operator to them can be determined analytically. But no linear combinations of these functions reproduce polynomials, such that approximations do not converge, in general. However, one can control the basic functions such that for any prescribed accuracy there exist simple approximants providing high order approximation rates up to this accuracy. This concept of approximate approximations will be discussed for approximations on uniform and nonuniform grids. Applications to the derivation of semi--analytic cubature formulas for multivariate integral operators and the solution of partial differential problems are given.