Convergence of repeated-pole rational approximation of the exponential on infinite sectors
Abstract
For h>0, degree m≥0, 0≤β<π, and q>0, let Σβ={reiθ:r≥0, |θ|≤β/2} and let Em,β(q) denote the best uniform error for approximating e−hz on Σβ by rational functions whose only possible finite pole is s=−q/h, with multiplicity at most m. For the linear scaling q=αm with a fixed α>0, a Möbius map reduces the problem to polynomial approximation on a compact lens. Write γβ=2−β/π and let αc(γβ) denote the saddle-coalescence threshold. For 0<α<αc(γβ), let Hβ(α)∈(0,1) be the exponential of the real saddle action. An exact boundary Faber integral, a global fractional-power contour deformation, and a two-saddle expansion yield the sharp two-sided order Em,β(αm)asymp m−1/2Hβ(α)m, uniformly on compact subsets of 0<α<αc(γβ). On this noncoalescing interval, the exponential factor Hβ(α) has the unique minimizer αβ*=1/[γβsin(π/(2γβ))] and minimum tan(π/(4γβ)), recovering the classical half-line factor tan(π/8)=√2−1 at β=0. Numerical-range estimates give matrix dimension-independent upper bounds for projected shift-and-invert Arnoldi, while finite-degree pole searches and nonnormal experiments illustrate the distinction between the sector-uniform law and individual matrix computations.
Related
- Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential
- Neural Approximation by Function Composition: Rigidity and Doubly Exponential Convergence
- Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching
- Exponential Consensus and Flocking in Multi-Agent Systems with Infinite Fading Memory
- Lower bounds in H2-rational approximation to Blaschke products
BibTeX
@misc{xue2026convergence,
title = {{Convergence of repeated-pole rational approximation of the exponential on infinite sectors}},
author = {Fei Xue and Tianqi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09251},
eprint = {2610.09251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09251},
}
RIS
TY - UNPB
TI - Convergence of repeated-pole rational approximation of the exponential on infinite sectors
AU - Fei Xue
AU - Tianqi Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09251
UR - https://arxiv.org/abs/2610.09251
AB - For h>0, degree m≥0, 0≤β<π, and q>0, let Σ_β={re^iθ:r≥0, |θ|≤β/2} and let E_m,β(q) denote the best uniform error for approximating e^−hz on Σ_β by rational functions whose only possible finite pole is s=−q/h, with multiplicity at most m. For the linear scaling q=αm with a fixed α>0, a Möbius map reduces the problem to polynomial approximation on a compact lens. Write γ_β=2−β/π and let α_c(γ_β) denote the saddle-coalescence threshold. For 0<α<α_c(γ_β), let H_β(α)∈(0,1) be the exponential of the real saddle action. An exact boundary Faber integral, a global fractional-power contour deformation, and a two-saddle expansion yield the sharp two-sided order E_m,β(αm)asymp m^−1/2H_β(α)^m, uniformly on compact subsets of 0<α<α_c(γ_β). On this noncoalescing interval, the exponential factor H_β(α) has the unique minimizer α_β^*=1/[γ_βsin(π/(2γ_β))] and minimum tan(π/(4γ_β)), recovering the classical half-line factor tan(π/8)=√2−1 at β=0. Numerical-range estimates give matrix dimension-independent upper bounds for projected shift-and-invert Arnoldi, while finite-degree pole searches and nonnormal experiments illustrate the distinction between the sector-uniform law and individual matrix computations.
ER -