  We consider the problem of approximating $[0,1]^{d}$-periodic functions by
convolution with a scaled Gaussian kernel. We start by establishing convergence
rates to functions from periodic Sobolev spaces and we show that the saturation
rate is $O(h^{2}),$ where $h$ is the scale of the Gaussian kernel. Taken from a
discrete point of view, this result can be interpreted as the accuracy that can
be achieved on the uniform grid with spacing $h.$ In the discrete setting, the
curse of dimensionality would place severe restrictions on the computation of
the approximation. For instance, a spacing of $2^{-n}$ would provide an
approximation converging at a rate of $O(2^{-2n})$ but would require
$(2^{n}+1)^{d}$ grid points. To overcome this we introduce a sparse grid
version of Gaussian convolution approximation, where substantially fewer grid
points are required, and
  show that the sparse grid version delivers a saturation rate of
$O(n^{d-1}2^{-2n}).$ This rate is in line with what one would expect in the
sparse grid setting (where the full grid error only deteriorates by a factor of
order $n^{d-1}$) however the analysis that leads to the result is novel in that
it draws on results from the theory of special functions and key observations
regarding the form of certain weighted geometric sums.
