Approximation by short exponential sums with geometric error decay based on Gauss quadrature
Abstract
We present new short exponential sum approximations of length N for f1(x)=1/(a+x) with a>0 on [0, ∞) and for f2(x)= e−x2/2σ with σ>0 on ℝ with geometric error decay ρ−2N for user-defined N ≥ 2 and ρ>1. The approximations are built over consecutive intervals [bj, bj+1) ⊂ [0, ∞), j ∈ ℕ0, with interval lengths that depend on ρ and grow exponentially for f1 and are equidistant for f2. All parameters determining the exponential sum approximations on [bj, bj+1) are easily computed from the initial parameters on [b0, b1), ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of f1 and f2. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of log(x) on [1,∞) and of the error function erf(x) with predictable geometric error decay. Numerical examples for N=8 and N=10 clearly illustrate the theoretical error estimates.
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BibTeX
@misc{plonka2026approximation,
title = {{Approximation by short exponential sums with geometric error decay based on Gauss quadrature}},
author = {Gerlind Plonka and Yannick Riebe and Annie Cuyt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.03855},
eprint = {2606.03855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.03855},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximation by short exponential sums with geometric error decay based on Gauss quadrature
AU - Gerlind Plonka
AU - Yannick Riebe
AU - Annie Cuyt
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2606.03855
UR - https://arxiv.org/abs/2606.03855
AB - We present new short exponential sum approximations of length N for f_1(x)=1/(a+x) with a>0 on [0, ∞) and for f_2(x)= e^−x^2/2σ with σ>0 on ℝ with geometric error decay ρ^−2N for user-defined N ≥ 2 and ρ>1. The approximations are built over consecutive intervals [b_j, b_j+1) ⊂ [0, ∞), j ∈ ℕ_0, with interval lengths that depend on ρ and grow exponentially for f_1 and are equidistant for f_2. All parameters determining the exponential sum approximations on [b_j, b_j+1) are easily computed from the initial parameters on [b_0, b_1), ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of f_1 and f_2. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of log(x) on [1,∞) and of the error function erf(x) with predictable geometric error decay. Numerical examples for N=8 and N=10 clearly illustrate the theoretical error estimates.
ER -