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 $L^\infty$ norm of functions on the sphere with bounded Laplacian whose first spherical harmonic components vanish. In every dimension $d\geq2$, these constants are expressed through weighted $L^1$ extremal polynomials. On $S^2$ the constant is an