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

Voir la notice du chapitre de livre

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. 
@article{TIMM_2012_18_4_a18,
     author = {S. V. Kuznetsov and K. S. Tikhanovtseva},
     title = {Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a~closed interval},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {211--223},
     year = {2012},
     volume = {18},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a18/}
}
TY  - JOUR
AU  - S. V. Kuznetsov
AU  - K. S. Tikhanovtseva
TI  - Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a closed interval
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2012
SP  - 211
EP  - 223
VL  - 18
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a18/
LA  - ru
ID  - TIMM_2012_18_4_a18
ER  - 
%0 Journal Article
%A S. V. Kuznetsov
%A K. S. Tikhanovtseva
%T Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a closed interval
%J Trudy Instituta matematiki i mehaniki
%D 2012
%P 211-223
%V 18
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a18/
%G ru
%F TIMM_2012_18_4_a18
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/

[1] Arestov V. V., Raevskaya V. Yu., “Odna ekstremalnaya zadacha dlya algebraicheskikh mnogochlenov s nulevym srednim znacheniem na otrezke”, Mat. zametki, 62:3 (1997), 332–342 | DOI | MR | Zbl

[2] Babenko A. G., Ekstremalnye svoistva polinomov i tochnye otsenki srednekvadratichnykh priblizhenii, Dis. kand. fiz.-mat. nauk, Sverdlovsk, 1987, 109 pp. | MR

[3] Krylov V. I., Priblizhennoe vychislenie integralov, Fizmatgiz, M., 1959, 327 pp. | MR

[4] Tikhanovtseva K. S., “O naimenshei mere mnozhestva neotritsatelnosti algebraicheskogo mnogochlena s nulevym vzveshennym srednim znacheniem na otrezke”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16, no. 4, 2010, 300–311