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 generalise and extend the differential approximation method proposed in [4, 40] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$ in the weighted 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 $\min\limits_{\lambda_{1},\ldots, \lambda_{N}, \gamma_{1}, \ldots, \gamma_{N}}\|\mathrm{e}^{-\cdot^{2}/2\sigma} - \sum_{j=1}^{N} \gamma_{j} \, {\mathrm e}^{\lambda_{j} \cdot}\|_{L^{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})}$, are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of ${\mathcal O}(N^{3})$ operations. 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. Our new convergence result for approximation of Gaussian functions by exponential sums of length $N$ shows that exponential error decay rates $e^{-cN}$ are not only achievable for complete monotone functions.