Differential approximation of the Gaussian by short cosine sums with exponential error decay

In this paper we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We extend the differential approximation method proposed in [4,39] to approximate $\mathrm{e}^{{-t{2}/2\sigma}$} in the weighted space $L2({\mathbb R}, \mathrm{e}^{{-t{2}/2\rho})$} where $\sigma, \, \rho >0$. We prove that the optimal frequency parameters $\lambda1, \ldots , \lambda{N}$ for this method in the approximation problem $ \min\limits{\lambda{1},\ldots, \lambda{N}, \gamma{1} \ldots \gamma{N}}|\mathrm{e}^{{-\cdot{2}/2\sigma}} - \sum\limits{j=1}^{N} \gamma{j} \, {\mathrm e}^{\lambda_{j} \cdot}|{L{2}({\mathbb R}, \mathrm{e}^{{-t{2}/2\rho})}$,} are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of $\mathit{O}(N^{3})$ operations. Furthermore, we derive a direct algorithm to solve this approximation problem based on a matrix pencil method for a special structured matrix. The entries of this matrix are determined by hypergeometric functions. For the weighted $L_{2}$-norm, we prove that the approximation error decays exponentially with respect to the length $N$ of the sum. An exponentially decaying error in the (unweighted) $L^{2}$-norm is achieved using a truncated cosine sum.