An Extremal Property of Chebyshev Polynomials

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 $k\ge 1$.