Geodesic
The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 264-270

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Let $\mathcal P_n(\alpha)$ be the set of algebraic polynomials $p_n$ of order $n$ with real coefficients and zero weighted mean value with ultraspherical weight $\varphi^{(\alpha)}(t)=(1-t^2)^\alpha$ on the interval $[-1,1]$: $\int_{-1}^1\varphi^{(\alpha)}(t)p_n(t)\,dx=0$. We study the problem on the smallest value $\mu_n=\inf\{m(p_n)\colon p_n\in\mathcal P_n(\alpha)\}$ of the weighted measure $m(p_n)=\int_{\mathcal X(p_n)}\varphi^{(\alpha)}(t)\,dt$ of the set where $p_n$ is nonnegative. The order of $\mu_n$ with respect to $n$ is found: it is proved that $\mu_n(\alpha)\asymp n^{-2(\alpha+1)}$ as $n\to\infty$.
Keywords: algebraic polynomials, polynomials with zero weighted mean value, ultraspherical weight. 
K. S. Tikhanovtseva. The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 264-270. http://geodesic.mathdoc.fr/item/TIMM_2014_20_1_a25/
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