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# On the convergence and saturation problem of a sequence of discrete linear operators of exponential type inL p(−∞, ∞) spaces

Acta Mathematica Hungarica
Author: D. Dryanov
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# Refinement of an inequality of P. L. Chebyshev

Acta Mathematica Hungarica
Authors: D. Dryanov and R. Fournier

## Abstract

Let P n denote the linear space of polynomials p(z:=Σk=0 n a k(p)z k of degree ≦ n with complex coefficients and let |p|[−1,1]: = maxx∈[−1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval [−1, 1]. Let t nP n be the n th Chebyshev polynomial. The inequality

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{\left| p \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (t_n )} \right|}},p \in P_n$$ \end{document}
due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t n in P n. The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.

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