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 Trudy Mat. Inst. Steklova, 2005, Volume 248, Pages 237–249 (Mi tm134)

An Extremal Property of Chebyshev Polynomials

V. D. Stepanov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: For any integer $k\ge 1$, in the metric of weighted classes $L^2(\omega )$, sharp two-sided inequalities of the form $\gamma _k|\int G^{(k)}(x) \nu _k(x) dx|^2\le [\mathrm {dist}_{L^2(\omega )}(G,\mathcal P_{k-1})]^2\le \gamma _k\int |G^{(k)}(x)|^2\nu _k(x) dx$ 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 $k\ge 1$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 248, 230–242

Bibliographic databases:
UDC: 517.51

Citation: V. D. Stepanov, “An Extremal Property of Chebyshev Polynomials”, Studies on function theory and differential equations, Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii, Trudy Mat. Inst. Steklova, 248, Nauka, MAIK «Nauka/Inteperiodika», M., 2005, 237–249; Proc. Steklov Inst. Math., 248 (2005), 230–242

Citation in format AMSBIB
\Bibitem{Ste05} \by V.~D.~Stepanov \paper An Extremal Property of Chebyshev Polynomials \inbook Studies on function theory and differential equations \bookinfo Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii \serial Trudy Mat. Inst. Steklova \yr 2005 \vol 248 \pages 237--249 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm134} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2165931} \zmath{https://zbmath.org/?q=an:1125.41306} \transl \jour Proc. Steklov Inst. Math. \yr 2005 \vol 248 \pages 230--242 

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This publication is cited in the following articles:
1. Stepanov V.D., “An extremal property of Jacobi polynomials in two-sided Chernoff-type inequalities for higher order derivatives”, Proc. Amer. Math. Soc., 136:5 (2008), 1589–1597
2. Smarzewski R., Rutka P., “Inequalities of Chernoff type for finite and infinite sequences of classical orthogonal polynomials”, Proc. Amer. Math. Soc., 138:4 (2010), 1305–1315
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