Differential approximation of the Gaussian by short cosine sums with exponential error decay
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In this paper we propose a method to approximate the Gaussian function on ℝ by a short cosine sum. We extend the differential approximation method proposed in [4,39] to approximate e^-t^2/2σ in the weighted space L_2(ℝ, e^-t^2/2ρ) where σ, ρ >0. We prove that the optimal frequency parameters λ_1, … , λ_N for this method in the approximation problem min_λ_1,…, λ_N, γ_1…γ_Ne^-·^2/2σ - ∑_j=1^Nγ_j e^λ_j·_L_2(ℝ, e^-t^2/2ρ), are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of 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.
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