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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Some remarks on spherical harmonics

Author: V. M. Gichev
Translated by: the author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 553-567
MSC (2000): Primary 33E30
Published electronically: June 1, 2009
MathSciNet review: 2473744
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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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Additional Information

V. M. Gichev
Affiliation: Omsk Division, Sobolev Mathematical Institute, Siberian Branch, Russian Academy of Sciences, Ul. Pevtsova 13, 644099 Omsk, Russia

Keywords: Nodal set, spherical harmonics, Hausdorff measure
Received by editor(s): September 11, 2007
Published electronically: June 1, 2009
Additional Notes: Supported in part by RFBR (grant nos. 06008-01403 and 06007-8951), and also by Sibirean Department of RAS (project no. 117)
Article copyright: © Copyright 2009 American Mathematical Society