Geodesic
On the least measure of the nonnegativity set of an algebraic polynomial with zero weighted mean value on a segment
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 300-311

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Let $\mathcal P_n(\varphi^{(\alpha)})$ be the set of algebraic polynomials $P_n$ of order $n$ with real coefficients and zero weighted mean value with respect to the ultraspherical weight $\varphi^{(\alpha)}(x)=(1-x^2)^\alpha$ on the interval $[-1,1]$: $\int_{-1}^1\varphi^{(\alpha)} P_n(x)\,dx=0$. We study the problem about the least possible value $\inf\{\mu(P_n)\colon P_n\in\mathcal P_n(\varphi^{(\alpha)})\}$ of the measure $\mu(P_n)=\int_{\mathcal X(P_n)}\varphi^{(\alpha)}(t)\,dt$ of the set $\mathcal X(P_n)=\{x\in[-1,1]\colon P_n(x)\ge0\}$ of points of the interval at which the polynomial $P_n\in\mathcal P_n(\varphi^{(\alpha)})$ is nonnegative. In this paper, the problem is solved for $n=2$ and $\alpha>0$. V. V. Arestov and V. Yu. Raevskaya solved the problem for $\alpha=0$ in 1997; in this case, an extremal polynomial has one interval of nonnegativity such that one of its endpoints coincides with one of the endpoints of the interval. In the case $\alpha>0$, we find that an extremal polynomial has two intervals of nonnegativity with endpoints $\pm1$.
Keywords: extremal problem, algebraic polynomials, polynomials with zero weighted mean value, ultraspherical weight. 
K. S. Tikhanovtseva. On the least measure of the nonnegativity set of an algebraic polynomial with zero weighted mean value on a segment. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 300-311. http://geodesic.mathdoc.fr/item/TIMM_2010_16_4_a28/
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