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 Mat. Zametki: Year: Volume: Issue: Page: Find

 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

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}