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In this paper we propose a method to approximate the Gaussian function on${\mathbb R}$ by a short cosine sum. We extend the differential approximationmethod proposed in [4,39] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$ in theweighted space $L_2({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})$ where $\sigma, \,\rho >0$. We prove that the optimal frequency parameters $\lambda_1, \ldots ,\lambda_{N}$ 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. Thisobservation leads us to a numerically stable approximation method with lowcomputational cost of $\mathit{O}(N^{3})$ operations. Furthermore, we derive adirect algorithm to solve this approximation problem based on a matrix pencilmethod for a special structured matrix. The entries of this matrix aredetermined by hypergeometric functions.  For the weighted $L_{2}$-norm, we prove that the approximation error decaysexponentially with respect to the length $N$ of the sum. An exponentiallydecaying error in the (unweighted) $L^{2}$-norm is achieved using a truncatedcosine sum.

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