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Mat. Zametki, 1972, Volume 12, Issue 5, Pages 501–510 (Mi mz9909)  

The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric

V. M. Veselinov

M. V. Lomonosov Moscow State University

Abstract: We investigate the approximation of functions by Bernstein polynomials. We prove that
$$ r_{[0,1]}(f, B_n(f))\leqslant\mu_f(4\sqrt{\frac{\ln n}{n}})+O(\sqrt{\frac{\ln n}{n}}),\eqno{(1)} $$
where $r_{[0,1]}(f, B_n(f))$ is the Hausdorff distance between the functions $f(x)$ and $B_n(f; x)$ in $[0,1]$,
$$ \mu_f(\delta)=\frac12\sup_{\substack{|x_1-x_2|\leqslant\deltax_1,x_2\in\Delta}}\{\sup_{x_1\leqslant x\leqslant x_2}[|f(x_1)-f(x)|+|f(x_2)-f(x)|]-|f(x_1)-f(x_2)|\} $$
is the modulus of nonmonotonicity of $f(x)$. The bound (1) is of better order than that obtained by Sendov. We show that the order of (1) cannot be improved.

Full text: PDF file (796 kB)

English version:
Mathematical Notes, 1972, 12:5, 737–742

Bibliographic databases:

UDC: 517.5
Received: 06.04.1972

Citation: V. M. Veselinov, “The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric”, Mat. Zametki, 12:5 (1972), 501–510; Math. Notes, 12:5 (1972), 737–742

Citation in format AMSBIB
\Bibitem{Ves72}
\by V.~M.~Veselinov
\paper The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric
\jour Mat. Zametki
\yr 1972
\vol 12
\issue 5
\pages 501--510
\mathnet{http://mi.mathnet.ru/mz9909}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=320592}
\zmath{https://zbmath.org/?q=an:0252.41003}
\transl
\jour Math. Notes
\yr 1972
\vol 12
\issue 5
\pages 737--742
\crossref{https://doi.org/10.1007/BF01099055}


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