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Positive values of harmonic polynomials. (English. Russian original) Zbl 1124.41026

Function spaces, approximations, and differential equations. Collected papers dedicated to Oleg Vladimirovich Besov on his 70th birthday. Transl. from the Russian. Moscow: Maik Nauka/Interperiodika. Proceedings of the Steklov Institute of Mathematics 243, 39-45 (2003); translation from Tr. Mat. Inst. Steklova 243, 46-52 (2003).
Summary: It is proved that, among all second-order spherical harmonics \(Y_2\), the quantity \(\text{meas}\{x\in S^2:Y_2(x)\geq 0\}\) attains its minimal value at a zonal polynomial. For harmonics of higher even orders, the situation is different. Several examples are considered.
For the entire collection see [Zbl 1064.46002].

MSC:

41A55 Approximate quadratures
42A05 Trigonometric polynomials, inequalities, extremal problems
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