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 $L_2({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})$ 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. 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.

翻译：本文提出一种利用短余弦和逼近实数域上高斯函数的方法。我们将文献[4,39]提出的微分近似方法推广至加权空间 $L_2({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})$ 中的函数 $\mathrm{e}^{-t^{2}/2\sigma}$ 逼近，其中 $\sigma, \, \rho >0$。研究表明，在逼近问题 $\min\limits_{\lambda_{1},\ldots, \lambda_{N}, \gamma_{1} \ldots \gamma_{N}}\|\mathrm{e}^{-\cdot^{2}/2\sigma} - \sum\limits_{j=1}^{N} \gamma_{j} \, {\mathrm e}^{\lambda_{j} \cdot}\|_{L_{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})}$ 中，该方法的最优频率参数 $\lambda_1, \ldots , \lambda_{N}$ 是尺度化Hermite多项式的零点。这一发现引导我们建立了一种数值稳定且计算复杂度仅为 $\mathit{O}(N^{3})$ 的逼近方法。进一步地，我们基于特殊结构化矩阵的矩阵束方法，推导了该逼近问题的直接求解算法。该矩阵的矩阵元由超几何函数确定。对于加权 $L_{2}$ 范数，我们证明了逼近误差随余弦和长度 $N$ 呈指数衰减。通过采用截断余弦和，可实现非加权 $L^{2}$ 范数下的指数误差衰减。