Geodesic
An extremal problem for algebraic polynomials with zero mean value on an interval
Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 332-342 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\mathscr P_n^0(h)$ be the set of algebraic polynomials of degree $n$ with real coefficients and with zero mean value (with weight $h$) on the interval $[-1,1]$: $$ \int_{-1}^1h(x)p_n(x)dx=0; $$ here $h$ is a function which is summable, nonnegative, and nonzero on a set of positive measure on $[-1,1]$. We study the problem of the least possible value $$ i_n(h)=\inf\{\mu(p_n):p_n\in\mathscr P_n^0\} $$ of the measure $\mu(p_n)=\operatorname{mes}\{x\in[-1,1]:p_n(x)\ge0\}$ of the set of points of the interval at which the polynomial $p_n\in\mathscr P_n^0$ is nonnegative. We find the exact value of $i_n(h)$ under certain restrictions on the weight $h$. In particular, the Jacobi weight $$ h^{(\alpha,\beta)}(x)=(1-x)^\alpha(1+x)^\beta $$ satisfies these restrictions provided that $-1<\alpha,\beta\le0$. 
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     author = {V. V. Arestov and V. Yu. Raevskaya},
     title = {An extremal problem for algebraic polynomials with zero mean value on an interval},
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}
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V. V. Arestov; V. Yu. Raevskaya. An extremal problem for algebraic polynomials with zero mean value on an interval. Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 332-342. http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a1/

[1] Babenko A. G., Ekstremalnye svoistva polinomov i tochnye otsenki srednekvadratichnykh priblizhenii, Diss. ... k. f.-m. n., Sverdlovsk, 1987

[2] Bernshtein S. N., “O formulakh kvadratur Kotesa i Chebysheva”, Sobranie sochinenii (v 4-kh tomakh), T. 2, M., 1954, 200–204

[3] Babenko A. G., “Ob odnoi ekstremalnoi zadache dlya polinomov”, Matem. zametki, 35:3 (1984), 349–356 | MR | Zbl

[4] Taikov L. V., “Odin krug ekstremalnykh zadach dlya trigonometricheskikh polinomov”, UMN, 20:3 (1965), 205–211 | MR | Zbl

[5] Babenko A. G., “Neravenstvo Dzheksona dlya srednekvadratichnykh priblizhenii periodicheskikh funktsii trigonometricheskimi polinomami na ravnomernoi setke”, Matem. zametki, 43:4 (1988), 460–473 | MR

[6] Fazekas G., Levinstein V. I., “On upper bounds for code distance and covering radius of designs in polynomial metric spaces”, J. Combin. Theory. Ser. A, 70:2 (1995), 267–288 | DOI | MR | Zbl

[7] Yudin V. A., “Pokrytiya sfery i ekstremalnye svoistva ortogonalnykh mnogochlenov”, Diskretnaya matem., 7:3 (1995), 81–88 | MR | Zbl

[8] Krylov V. I., Priblizhennoe vychislenie integralov, Fizmatgiz, M., 1959

[9] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962