Logarithmic controlled bandwidth for locally analytic functions
Let $f : [-1,1] \to \mathbb{C}$ be nonzero with a holomorphic extension to an open complex neighbourhood of $[-1,1]$, and set $M = \|f\|_{L^\infty([-1,1])}$. Use the Fourier convention $$\widehat g(\xi) = \frac{1}{2\pi} \int_{\mathbb{R}} g(x) e^{-i\xi x}\,dx.$$ For $0 < \varepsilon < 1$ let $c_f(\varepsilon)$ be the infimum of the $c > 0$ for which there is a Schwartz function $g \in \mathcal{S}(\mathbb{R})$ with $$\operatorname{supp} \widehat g \subset [-c, c], \qquad \|f - g\|_{L^\infty([-1,1])} \le \varepsilon M,$$ $$\|g\|_{L^\infty(\mathbb{R})} \le 4M, \qquad \|\widehat g\|_{L^\infty(\mathbb{R})} \le 4M, \qquad \|(\widehat g)'\|_{L^\infty(\mathbb{R})} \le 4M.$$ Chen, Serkh and Bremer, who use such approximations to bound the error of the adaptive Levin method for oscillatory integrals, showed that $c_f(\varepsilon) = O(\varepsilon^{-1/m})$ as $\varepsilon \to 0$, for every positive integer $m$, when $f$ has an infinitely differentiable extension to a neighbourhood of $[-1,1]$, and noted that $c_f(\varepsilon) = O(\log(1/\varepsilon))$ when $f$ extends to a bounded analytic function on a horizontal strip containing the real axis.
Problem. Are there constants $C_f < \infty$ and $\varepsilon_f > 0$, which may depend on $f$ and its neighbourhood of analyticity but not on $\varepsilon$, such that $$c_f(\varepsilon) \le C_f \log(1/\varepsilon) \qquad (0 < \varepsilon < \varepsilon_f)?$$ All three global norm bounds are part of the problem.
Source: Chen, Serkh and Bremer (2024), §2.3, the conjecture after (42), for c_f(ε) as in their Definition 1; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, bandlimited approximation, Levin method, oscillatory integrals
References
@article{chen2024,
author = {Chen, Shukui and Serkh, Kirill and Bremer, James},
title = {On the adaptive {L}evin method},
journal = {Numer. Math.},
volume = {156},
number = {6},
year = {2024},
pages = {1927--1985},
doi = {10.1007/s00211-024-01443-6},
eprint = {2211.13400},
archivePrefix = {arXiv},
}
@misc{aim-analytic-controlled-bandwidth,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Logarithmic controlled bandwidth for locally analytic functions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/664-analytic-controlled-bandwidth.md},
note = {Entry 664 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
Yes: for every nonzero f analytic in a neighbourhood of [-1,1], c_f(epsilon) <= C_f log(1/epsilon) for small epsilon, with all three global norm bounds, by an epsilon-dependent smooth cutoff and a smooth Fourier truncation..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 593-analytic-bandwidth/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00662, "Logarithmic controlled bandwidth for locally analytic functions", clopen.solutions.