On best approximation and estimates of Bohr--Favard type on the sphere for functions with bounded Laplacian
Abstract
We determine the best constants in estimates of the L∞ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension d≥2, these constants are expressed through weighted L1 extremal polynomials. On S2 the constant is an explicit positive series, and the best coefficient of n−2 is 7ζ(3)/4. On S3 the best coefficient of n−2 is a positive integral involving the Hörmander--Bernhardsson extremal function. In higher dimensions the rescaled extremal polynomials converge to weighted entire extremals, which give asymptotic formulas for the Bohr constants on S4 and S6. The most unexpected and, in our view, interesting observation is that, for every fixed n≥2 on S2 and S3, the constants in best approximation estimates are \emph{strictly smaller} than the corresponding Bohr--Favard constants.
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BibTeX
@misc{solyanik2026best,
title = {{On best approximation and estimates of Bohr--Favard type on the sphere for functions with bounded Laplacian}},
author = {Alexey Solyanik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14811},
eprint = {2610.14811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On best approximation and estimates of Bohr--Favard type on the sphere for functions with bounded Laplacian
AU - Alexey Solyanik
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14811
UR - https://arxiv.org/abs/2610.14811
AB - We determine the best constants in estimates of the L^∞ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension d≥2, these constants are expressed through weighted L^1 extremal polynomials. On S^2 the constant is an explicit positive series, and the best coefficient of n^−2 is 7ζ(3)/4. On S^3 the best coefficient of n^−2 is a positive integral involving the Hörmander--Bernhardsson extremal function. In higher dimensions the rescaled extremal polynomials converge to weighted entire extremals, which give asymptotic formulas for the Bohr constants on S^4 and S^6. The most unexpected and, in our view, interesting observation is that, for every fixed n≥2 on S^2 and S^3, the constants in best approximation estimates are \emph{strictly smaller} than the corresponding Bohr--Favard constants.
ER -