Geodesic
Mean asymmetry of polynomials on compact homogeneous spaces
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 566-582.

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Let $M=G/H$ be a homogeneous space of a compact Lie group $G$, ${{\mathcal E}}$ be an $G$-invariant finite dimensional subspace of $L^2_{\mathord{\mathbb{R}}}(M)$, and ${\mathord{\mathcal{S}}}$ be the unit sphere in it. Set $\eta_a(u)=\int_M\left(u_+^a(x)-u_-^a(x)\right)\,dx$, where $u_+(x)=\max\{u(x),0\}$, $u_-(x)=-\min\{u(x),0\}$. We consider the asymptotic behavior of the variance of the random variable $\eta_a$ as $a\to\infty$ or $\dim{{\mathcal E}}\to\infty$ for the uniform distribution of $u$ in ${\mathord{\mathcal{S}}}$. For instance, if ${{\mathord{\mathcal{E}}}}$ is the space of trigonometrical polynomials of degree less or equal to $n$, then $\mathop{\mathrm{Var}}(\eta_a)\sim \frac{A}{n}$ as $n\to\infty$.
Keywords: compact homogeneous space, sums of Laplace–Beltrami eigenfunctions, defect of symmetry. 
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V. M. Gichev. Mean asymmetry of polynomials on compact homogeneous spaces. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 566-582. http://geodesic.mathdoc.fr/item/SEMR_2013_10_a31/

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