Some remarks on spherical harmonics

Gichev, V.

doi:10.1090/S1061-0022-09-01061-9

Some remarks on spherical harmonics
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by V. M. Gichev
Translated by: the author

St. Petersburg Math. J. 20 (2009), 553-567

DOI: https://doi.org/10.1090/S1061-0022-09-01061-9

Published electronically: June 1, 2009

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Abstract:

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).

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