Some remarks on spherical harmonics

Abstract

The article contains several observations on spherical harmonics and their nodal sets: a construction for harmonics with prescribed zeroes; a kind of canonical representation of this type for harmonics on bbS^2; upper and lower bounds for nodal length and inner radius (the upper bounds are sharp); precise upper bound for the number of common zeroes of two spherical harmonics on bbS^2; the mean Hausdorff measure on the intersection of k nodal sets for harmonics of different degrees on bbS^m, where kleq m (in particular, the mean number of common zeroes of m harmonics).