Convergence of multilevel stationary Gaussian convolution

In this paper we give a short note showing convergence rates for periodic approximation of smooth functions by multilevel Gaussian convolution. We will use the Gaussian scaling in the convolution at the finest level as a proxy for degrees of freedom d in the model. We will show that, for functions in the native space of the Gaussian, convergence is of the order (Formula Presented.). This paper provides a baseline for what should be expected in discrete convolution, which will be the subject of a follow up paper.