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

Nadiia Derevianko, Gerlind Plonka

TL;DR

This work addresses efficiently approximating the Gaussian ${e^{-t^{2}/(2\sigma)}}$ on ${\mathbb R}$ by a short cosine sum in the weighted space ${L^{2}({\mathbb R}, e^{-t^{2}/(2\rho)})}$. It builds a differential-approximation framework where an optimal differential operator $D_N$ yields a characteristic polynomial $P_N(\lambda)$ whose zeros are precisely the optimal frequencies, which turn out to be zeros of a scaled Hermite polynomial; this leads to a numerically stable, ${\mathcal O}(N^{3})$-cost algorithm. The method provides exponential convergence in the weighted $L^{2}$-norm, with explicit rates $< c(\frac{r}{\sqrt{2(2r+1)}})^N N^{3/4}$ for $r=\rho/\sigma$, and similar exponential decay for a truncated unweighted norm. By comparing to Prony-type methods and ESPRIT/ESPIRA, the authors demonstrate superior stability and efficiency for obtaining real cosine-sum representations, and discuss extensions to finite intervals via Hermite-quadrature-based analysis. The results bridge Gaussian approximation, Hermite polynomials, and matrix-pencil techniques, offering a principled route to high-accuracy short-exponential-sum representations with potential applications in numerical analysis and signal processing.

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 $\mathrm{e}^{-t^{2}/2σ}$ in the weighted space $L^{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2ρ})$ where $σ, \, ρ>0$. We prove that the optimal frequency parameters $λ_1, \ldots , λ_{N}$ for this method in the approximation problem $ \min\limits_{λ_{1},\ldots, λ_{N}, γ_{1}, \ldots, γ_{N}}\|\mathrm{e}^{-\cdot^{2}/2σ} - \sum_{j=1}^{N} γ_{j} \, {\mathrm e}^{λ_{j} \cdot}\|_{L^{2}({\mathbb R}, \mathrm{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 ${\mathcal O}(N^{3})$ 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.

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

TL;DR

This work addresses efficiently approximating the Gaussian on by a short cosine sum in the weighted space . It builds a differential-approximation framework where an optimal differential operator yields a characteristic polynomial whose zeros are precisely the optimal frequencies, which turn out to be zeros of a scaled Hermite polynomial; this leads to a numerically stable, -cost algorithm. The method provides exponential convergence in the weighted -norm, with explicit rates for , and similar exponential decay for a truncated unweighted norm. By comparing to Prony-type methods and ESPRIT/ESPIRA, the authors demonstrate superior stability and efficiency for obtaining real cosine-sum representations, and discuss extensions to finite intervals via Hermite-quadrature-based analysis. The results bridge Gaussian approximation, Hermite polynomials, and matrix-pencil techniques, offering a principled route to high-accuracy short-exponential-sum representations with potential applications in numerical analysis and signal processing.

Abstract

In this paper, we propose a method to approximate the Gaussian function on by a short cosine sum. We generalise and extend the differential approximation method proposed in [4, 40] to approximate in the weighted space where . We prove that the optimal frequency parameters for this method in the approximation problem , are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of 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 -norm, we prove that the approximation error decays exponentially with respect to the length of the sum. An exponentially decaying error in the (unweighted) -norm is achieved using a truncated cosine sum. Our new convergence result for approximation of Gaussian functions by exponential sums of length shows that exponential error decay rates are not only achievable for complete monotone functions.
Paper Structure (14 sections, 5 theorems, 114 equations, 4 figures, 3 algorithms)

This paper contains 14 sections, 5 theorems, 114 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Differential approximation method
  3. Contribution of this paper
  4. Related results
  5. Differential Method for Approximation of the Gaussian on $\mathbb{R}$
  6. Approximation with frequency parameters being zeros of a scaled Hermite polynomial
  7. Differential approximation method as a matrix pencil method
  8. Error estimates for approximation of the Gauss function
  9. Error estimate in the weighted $L^{2}$-norm
  10. Error estimate in the $L^{2}({\mathbb R})$-norm
  11. Relation to other approximation algorithms
  12. Prony's method based on differential operator
  13. Comparison with ESPRIT and ESPIRA
  14. Conclusion

Key Result

Theorem 2.1

For $\rho >0$, the minimizing vecor ${\mathbf b} \in {\mathbb C}^{N}$ of the functional $F_\rho({\mathbf b})$ in $(Fbrho0)$ is given by ${\mathbf b}= ({b}_{0}, {b}_{1}, \ldots , {b}_{N-1})^{T}$ with Moreover, the corresponding characteristic polynomial $P_N(\lambda)$ is a weighted scaled Hermite polynomial of degree $N$,

Figures (4)

  • Figure 1: Decay of approximation error in logarithmic scale with respect to $N=1,\ldots,18$, computed with Algorithm \ref{['alg11']} for $\sigma=0.8$ with $\rho=1$ (left) and $\rho=2$ (right).
  • Figure 2: Approximation error in logarithmic scale with respect to $N=1,\ldots,18$ computed with Algorithm \ref{['alg11']} (blue points) and Algorithm \ref{['alg1']} (red diamonds) for $\sigma=0.8$ with $\rho=1$ (left) and $\sigma=1.25$ with $\rho=1.75$ (right).
  • Figure 3: Approximation error in logarithmic scale with respect to $N=1,\ldots,12$ computed with Algorithm \ref{['alg11']} (blue points) and Algorithm \ref{['alg3']} (red crosses) for $\sigma=0.8$ with $\rho=1$ (left) and $\sigma=1.25$ with $\rho=1.75$ (right).
  • Figure 4: The case $\sigma=1.25$, $\rho=\sigma/2$ and $N=16$ (cosine sum of lenght $8$). Left: Gaussian function, Middle: Error in $L_\infty([-5,5])$-norm for ESPRIT (blue), ESPIRA (red) and Algorithm \ref{['alg11']} (black), Right: error in $L_\infty([-5,5],\mathrm{e}^{-t^2/4\rho})$ for ESPRIT (blue), ESPIRA (red) and Algorithm \ref{['alg11']} (black).

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 5 more