Some remarks on spherical harmonics
arXiv:0705.2547
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 nodal sets for harmonics of different degrees on $\bbS^m$, where (in particular, the mean number of common zeroes of harmonics).
20 pages. Translation of the version which is published in Russian. Minor modifications throughout the paper. An error in calculation is corrected (page 14)