On best approximation and estimates of Bohr--Favard type on the sphere for functions with bounded Laplacian

We determine the best constants in estimates of the $L^{\mathrm{\infty }}$ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension $d\ge 2$, 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 $7\zeta (3)\mathrm{/}4$. 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 $n\ge 2$ on $S^2$ and $S^3$, the constants in best approximation estimates are \emph{strictly smaller} than the corresponding Bohr–Favard constants.