Differential approximation of the Gaussian by short cosine sums with exponential error decay
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909618889818112 |
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| author | Derevianko, Nadiia Plonka, Gerlind |
| author_facet | Derevianko, Nadiia Plonka, Gerlind |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13587 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differential approximation of the Gaussian by short cosine sums with exponential error decay Derevianko, Nadiia Plonka, Gerlind Numerical Analysis 41A20, 42A16, 42C15, 65D15, 94A12 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. |
| title | Differential approximation of the Gaussian by short cosine sums with exponential error decay |
| topic | Numerical Analysis 41A20, 42A16, 42C15, 65D15, 94A12 |
| url | https://arxiv.org/abs/2307.13587 |