Geodesic
Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a closed interval
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 211-223 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\varphi^{(\alpha)}(t)=(1-t^2)^\alpha$ on the interval $[-1,1]$: $\int_{-1}^1\varphi^{(\alpha)}(t)p_n(t)\,dx=0$. We study the problem on the smallest 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 $\mathcal X(p_n)=\{t\in[-1,1]\colon p_n(t)\ge0\}$ of points of the interval at which the polynomial $p_n\in\mathcal P_n(\varphi^{(\alpha)})$ is nonnegative. In this paper, the properties of an extremal polynomial of this problem are studied and an exact solution is presented for the case of cubic polynomials.
Keywords: algebraic polynomials, polynomials with zero weighted mean value, ultraspherical weight. 
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     title = {Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a~closed interval},
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S. V. Kuznetsov; K. S. Tikhanovtseva. Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a closed interval. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 211-223. http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a18/

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