Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential
Abstract
We study uniform approximation of exp(tz), t>0, on (−∞,0] by rational functions Pm(z)/(qm−z)m with a prescribed repeated pole qm>0. A Möbius transformation reduces the problem to polynomial approximation of Fλ(x)=exp (−λ(1−x)/(1+x)) on [−1,1], where λ=tqm. We derive a uniform two-saddle asymptotic formula for the Chebyshev coefficients, including explicit amplitude and phase and a relative remainder for each localized complex saddle contribution, covering fixed, sublinear, and linear pole scalings away from saddle coalescence. Because the two saddle contributions can cancel in a single coefficient, we pass to a growing block of neighboring coefficients and prove that the whole block cannot cancel. This transfers the coefficient asymptotics to approximation errors. For qm=(α/t)m, 0<α<3√3/2, the best uniform error has two-sided order m−1/2He(α)m; at the optimal ratio α=1/√2 this becomes m−1/2(√2−1)m. The normalized Chebyshev-weighted L2 projection error has an explicit bounded oscillatory profile. These prefactor-resolved estimates yield a two-term precision-to-work law and quantify mismatch between pole-design and stopping degrees. Finally, the scalar error gives an exact worst-case matrix-action benchmark for self-adjoint negative semidefinite matrices and dimension-independent shift-and-invert Krylov bounds.
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BibTeX
@misc{xue2026uniform,
title = {{Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential}},
author = {Fei Xue and Tianqi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21028},
eprint = {2609.21028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21028},
}
RIS
TY - UNPB
TI - Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential
AU - Fei Xue
AU - Tianqi Zhang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21028
UR - https://arxiv.org/abs/2609.21028
AB - We study uniform approximation of exp(tz), t>0, on (−∞,0] by rational functions P_m(z)/(q_m−z)^m with a prescribed repeated pole q_m>0. A Möbius transformation reduces the problem to polynomial approximation of F_λ(x)=exp (−λ(1−x)/(1+x)) on [−1,1], where λ=tq_m. We derive a uniform two-saddle asymptotic formula for the Chebyshev coefficients, including explicit amplitude and phase and a relative remainder for each localized complex saddle contribution, covering fixed, sublinear, and linear pole scalings away from saddle coalescence. Because the two saddle contributions can cancel in a single coefficient, we pass to a growing block of neighboring coefficients and prove that the whole block cannot cancel. This transfers the coefficient asymptotics to approximation errors. For q_m=(α/t)m, 0<α<3√3/2, the best uniform error has two-sided order m^−1/2H_e(α)^m; at the optimal ratio α=1/√2 this becomes m^−1/2(√2−1)^m. The normalized Chebyshev-weighted L^2 projection error has an explicit bounded oscillatory profile. These prefactor-resolved estimates yield a two-term precision-to-work law and quantify mismatch between pole-design and stopping degrees. Finally, the scalar error gives an exact worst-case matrix-action benchmark for self-adjoint negative semidefinite matrices and dimension-independent shift-and-invert Krylov bounds.
ER -