An Extremal Property of Chebyshev Polynomials

V. D. Stepanov

Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 237-249.

Résumé

For any integer $k\ge 1$, in the metric of weighted classes $L^2(\omega )$, sharp two-sided inequalities of the form $\gamma _k\bigl |\int G^{(k)}(x) \nu _k(x)\,dx\bigr |^2\le \bigl [\mathrm {dist}_{L^2(\omega )}(G,\mathcal P_{k-1})\bigr ]^2\le \gamma _k\int \bigl |G^{(k)}(x)\bigr |^2\nu _k(x)\,dx$ are obtained for the distance between an element $G$ and the subspace $\mathcal P_{k-1}$ of all polynomials of degree ${\le }\,k-1$; these inequalities reduce to equalities for Chebyshev-type polynomials of degree $k$. On the real axis with $\omega (x)=\nu _k(x)=\frac {1}{\sqrt {2\pi }}\,e^{-x^2/2}$ and $\gamma _k=1/k!$, a precise extension of the Chernoff inequality ($k=1$) is obtained for all $k\ge 1$.

Bibliographie

[1] Chernoff H., “A note on an inequality involving the normal distribution”, Ann. Probab., 9 (1981), 533–535 | DOI | MR | Zbl

[2] Suetin P.K., Klassicheskie ortogonalnye mnogochleny, 2-e izd., Nauka, M., 1979, 415 pp.

[3] Nikolskii S.M., Kvadraturnye formuly, 2-e izd., Nauka, M., 1974, 222 pp.

[4] Brascamp H., Lieb E., “On extensions of the Brunn–Minkowski and Prékopa–Leindler theorem, including inequalities for log concave functions, and with an application to the diffusion equation”, J. Funct. Anal., 22 (1976), 366–389 | DOI | MR | Zbl

[5] Bischoff W., Fichter M., “Optimal lower and upper bounds for the $L_p$-mean deviation of functions of a random variable”, Math. Meth. Stat., 9 (2000), 237–269 | MR | Zbl