Sharp asymptotic constant for rational approximation of fractional inverse powers
Fix $0 < a < b < \infty$ and $0 < \gamma < 1$, and let $$E_{n-1,n}(a, b; \gamma) = \inf_{p, q} \max_{x \in [a, b]} \left| x^{-\gamma} - \frac{p(x)}{q(x)} \right|,$$ over real polynomials $p, q$ with $\deg p \le n - 1$, $\deg q \le n$ and $q$ without zeros on $[a, b]$; poles outside $[a, b]$ are not restricted. Let $$k = \frac{\sqrt b - \sqrt a}{\sqrt b + \sqrt a}, \qquad K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}, \qquad \mu(k) = \frac{\pi K\bigl(\sqrt{1 - k^2}\bigr)}{2 K(k)}.$$ Braess and Hackbusch show that the best relative approximation of $\sqrt x$ on $[a, b]$ by rational functions of type $(n, n-1)$ has error between $4 e^{-2n\mu(k)}(1 - O(e^{-4n\mu(k)}))$ and $4 e^{-2n\mu(k)}$, and that $E_{n-1,n}(a, b; 1/2)$ lies between constant multiples of that error for large $n$; they note that, by a result of Gonchar, $E_{n-1,n}(a, b; \gamma)$ has the same asymptotic behaviour for every $0 < \gamma < 1$. Their numerical results led them to the conjecture below.
Problem. Prove or disprove the Braess–Hackbusch conjecture $$\lim_{n \to \infty} e^{2n\mu(k)} E_{n-1,n}(a, b; \gamma) = 4 (ab)^{-\gamma/2} \sin(\pi\gamma),$$ with the interval and the exponent fixed as $n \to \infty$.
Source: Braess and Hackbusch (2026), Conjecture 4.8.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, fractional powers, rational approximation, Stieltjes functions
References
@article{braess2026,
author = {Braess, Dietrich and Hackbusch, Wolfgang},
title = {On the approximation of {S}tieltjes functions by exponential sums and rational functions with applications to partial differential equations},
journal = {Numer. Math.},
volume = {158},
number = {2},
year = {2026},
pages = {455--490},
doi = {10.1007/s00211-025-01523-1},
note = {Revised preprint: Max Planck Institute for Mathematics in the Sciences, Leipzig, Preprint 10/2024, https://files-www.mis.mpg.de/mpi-typo3/preprints/2024/preprint2024_10.pdf},
}
@misc{aim-stieltjes-rational-asymptotic-constant,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp asymptotic constant for rational approximation of fractional inverse powers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/554-stieltjes-rational-asymptotic-constant.md},
note = {Entry 554 at commit a153f46, 2026-10-13},
}
Resolution claims
For a fixed positive finite interval and a fixed exponent between zero and one, we prove the Braess--Hackbusch conjecture for the leading constant in best absolute uniform rational approximation of a fractional inverse power. The approximants have numerator degree at most $n-1$ and denominator degree at most $n$, with no restriction on poles outside the interval. The proof combines strong …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00774, "Sharp asymptotic constant for rational approximation of fractional inverse powers", clopen.solutions.