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

TL;DR

This paper introduces a stable, efficient method to approximate Gaussian functions using short cosine sums with exponential error decay, extending differential approximation techniques and linking optimal parameters to Hermite polynomial zeros.

Contribution

It generalizes differential approximation for Gaussian functions, providing a numerically stable method with low computational cost and proven exponential error decay.

Findings

Optimal frequency parameters are zeros of scaled Hermite polynomials.

Approximation error decays exponentially with sum length N.

Method achieves exponential error decay in weighted and unweighted L2 norms.

Abstract

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.