Convex Bernstein limits and sharp recovery on expanding intervals
Abstract
We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most σ>0. For intervals of radius (n−γn)/σ, where 0≤γn<n and γn=o(n), one margin sequence gives this limit for every convex function of polynomial growth if and only if γn/[n1/3(log n)2/3]→∞. When this condition fails, a fixed uniformly strictly convex entire function has approximation errors with unbounded limsup. We apply the limit theorem to prove existence of the convex Bernstein constant for |x|λ for every λ≥1, and identify it with an attained entire-function minimum. We also give a sufficient margin for growth bounded by exp(a|x|α), 0<α<1. For arbitrary subexponential growth on the real line, each target with finite entire-function error admits a suitable margin, whereas no single margin is sufficient for the whole class. Further results concern attainment, right continuity with respect to the exponential type, weighted growth conditions, and the endpoints of fixed-fraction approximation intervals.
Related
- Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line
- Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
- Interchanging limits in the mutual information of two intervals separated by one site
- Sub-Gaussian Confidence Intervals for Heavy-Tailed Data: Characterizing the Limits of Inference
- Inverse limits of locally convex Lie groupoids
BibTeX
@misc{colbrook2026convex,
title = {{Convex Bernstein limits and sharp recovery on expanding intervals}},
author = {Matthew J. Colbrook and Catherine Drysdale},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31546},
eprint = {2610.31546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31546},
}
RIS
TY - UNPB
TI - Convex Bernstein limits and sharp recovery on expanding intervals
AU - Matthew J. Colbrook
AU - Catherine Drysdale
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31546
UR - https://arxiv.org/abs/2610.31546
AB - We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most σ>0. For intervals of radius (n−γ_n)/σ, where 0≤γ_n<n and γ_n=o(n), one margin sequence gives this limit for every convex function of polynomial growth if and only if γ_n/[n^1/3(log n)^2/3]→∞. When this condition fails, a fixed uniformly strictly convex entire function has approximation errors with unbounded limsup. We apply the limit theorem to prove existence of the convex Bernstein constant for |x|^λ for every λ≥1, and identify it with an attained entire-function minimum. We also give a sufficient margin for growth bounded by exp(a|x|^α), 0<α<1. For arbitrary subexponential growth on the real line, each target with finite entire-function error admits a suitable margin, whereas no single margin is sufficient for the whole class. Further results concern attainment, right continuity with respect to the exponential type, weighted growth conditions, and the endpoints of fixed-fraction approximation intervals.
ER -