Shifted tapered-pole approximation on finite sectors: an exact asymptotic and dimension control
Abstract
We study explicit lightning-plus-polynomial (LP) approximation of g(z)zα(log z)m on a closed unit-radius sector, where 0<α<1, m is fixed, and g is analytic nearby. Since the basic sector root-exponential rate is already known, we address three finer questions. First, a one-sided Abel--Poisson formula and a pole-free notched contour give a branchwise error bound that is uniform for grid shifts in compact subsets of (0,1) and valid for every fixed logarithmic power. Second, for zα in the dense and balanced regimes, an exact vertex-scale rescaling identifies a positive finite leading constant for a specified shifted sequence. Third, a rate-matched band of additional representation poles followed by polynomial compression gives the explicit representation dimension N+2√(2αλσN)+O(1) while leaving the first N tapered poles unchanged. A slightly overresolved hybrid sequence also preserves the exact leading constant. Matched-dimension experiments, including reentrant sectors, illustrate the allocation. These exact-asymptotic and dimension statements concern the stated constructions, not sector minimax approximation or total computational complexity.
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BibTeX
@misc{xue2026shifted,
title = {{Shifted tapered-pole approximation on finite sectors: an exact asymptotic and dimension control}},
author = {Fei Xue and Tianqi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09256},
eprint = {2610.09256},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09256},
}
RIS
TY - UNPB
TI - Shifted tapered-pole approximation on finite sectors: an exact asymptotic and dimension control
AU - Fei Xue
AU - Tianqi Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09256
UR - https://arxiv.org/abs/2610.09256
AB - We study explicit lightning-plus-polynomial (LP) approximation of g(z)z^α(log z)^m on a closed unit-radius sector, where 0<α<1, m is fixed, and g is analytic nearby. Since the basic sector root-exponential rate is already known, we address three finer questions. First, a one-sided Abel--Poisson formula and a pole-free notched contour give a branchwise error bound that is uniform for grid shifts in compact subsets of (0,1) and valid for every fixed logarithmic power. Second, for z^α in the dense and balanced regimes, an exact vertex-scale rescaling identifies a positive finite leading constant for a specified shifted sequence. Third, a rate-matched band of additional representation poles followed by polynomial compression gives the explicit representation dimension N+2√(2αλ_σN)+O(1) while leaving the first N tapered poles unchanged. A slightly overresolved hybrid sequence also preserves the exact leading constant. Matched-dimension experiments, including reentrant sectors, illustrate the allocation. These exact-asymptotic and dimension statements concern the stated constructions, not sector minimax approximation or total computational complexity.
ER -