Some remarks on spherical harmonics

Several observations on spherical harmonics and their nodal sets are presented: a construction for harmonics with prescribed zeros; a natural representation for harmonics on $\mathbb S^2$; upper and lower bounds for the nodal length and the inner radius (the upper bounds are sharp); the sharp upper bound for the number of common zeros of two spherical harmonics on $\mathbb S^2$; the mean Hausdorff measure of the intersection of $k$ nodal sets for harmonics of different degrees on $\mathbb S^m$, where $k\leq m$ (in particular, the mean number of common zeros of $m$ harmonics).