Geodesic
Positive Values of Harmonic Polynomials
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 46-52 
N. N. Andreev; V. A. Yudin. Positive Values of Harmonic Polynomials. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 46-52. http://geodesic.mathdoc.fr/item/TM_2003_243_a4/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is proved that, among all second-order spherical harmonics $Y_2$, the quantity $\mathrm {meas}\{x\in S^2\colon Y_2(x)\ge 0\}$ attains its minimal value at a zonal polynomial. For harmonics of higher even orders, the situation is different. Several examples are considered.

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[2] Armitage D. H., Gardiner S. J., “Best one-sided $L^1$-approximation by harmonic and subharmonic functions”, Advances in multivariate approximation, Proc. 3rd Intern. Conf. (Witten-Bommerholz, Germany, 1998), Math. Res., 107, eds. W. Haussmann, K. Jetter, M. Reimer, Wiley-VCH, Berlin, 1999, 43–56 | MR | Zbl

[3] Gobson E. V., Teoriya sfericheskikh i ellipsoidalnykh funktsii, Izd-vo inostr. lit., M., 1952