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

Abstract In this paper, we propose a method to approximate the Gaussian function on $$\mathbb{R}$$ by a short cosine sum. We generalize and extend the differential approximation method proposed in [6, 8] 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.