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 Mat. Zametki, 1977, Volume 22, Issue 1, Pages 85–101 (Mi mz8028)

The correctness problem for best approximations by trigonometric polynomials in the class $W_0^rH[\omega]_C$

A. V. Kroó

M. V. Lomonosov Moscow State University

Abstract: Suppose that $k$, $r\in Z_+$, $W_0^rH[\omega]_C=\{f:f is a$2\pi$-periodic function, f\in C^r[-\pi,\pi],\omega(f^{(r)},\delta)\le\omega(\delta)\}$, $T_k$ is the space of trigonometric polynomials of order $k$, $p_k(f)\in T_k$ is the polynomial of best uniform approximation to $f$, and $E_k(f)$ is the error of the best approximation. It is shown that for an arbitrary $\varepsilon>0$ we have,
\begin{gather*} \sup\limits_{f\in W_0^rH[\omega]_C}\sup\limits_{\substack{q_k\in T_k
\|f-q_k\|\le E_k(f)+\varepsilon}}\|p_k(f)-q_k\|_C\asymp R(\varepsilon),
\sup\limits_{f\in W_0^rH[\omega]_C}\sup\limits_{\substack{f_1\in C[-\pi,\pi]
\|f-f_1\|\le\varepsilon}}\|p_k(f)-p_k(f_1)\|_C\asymp R(\varepsilon), \end{gather*}
where for $0<\varepsilon\le\omega(1)$, $k>0$, $R(\varepsilon)$ is the root of the equation $R=(\varepsilon'R)^{r/(2k)}\omega((\varepsilon'R)^{1/(2k)})$, and for $k=0$ or $\varepsilon>\omega(1)$ we have $R(\varepsilon)=\varepsilon$.

Full text: PDF file (1034 kB)

English version:
Mathematical Notes, 1977, 22:1, 536–546

Bibliographic databases:

Citation: A. V. Kroó, “The correctness problem for best approximations by trigonometric polynomials in the class $W_0^rH[\omega]_C$”, Mat. Zametki, 22:1 (1977), 85–101; Math. Notes, 22:1 (1977), 536–546

Citation in format AMSBIB
\Bibitem{Kro77} \by A.~V.~Kro\'o \paper The correctness problem for best approximations by trigonometric polynomials in the class $W_0^rH[\omega]_C$ \jour Mat. Zametki \yr 1977 \vol 22 \issue 1 \pages 85--101 \mathnet{http://mi.mathnet.ru/mz8028} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=447935} \zmath{https://zbmath.org/?q=an:0361.42001} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 1 \pages 536--546 \crossref{https://doi.org/10.1007/BF01147696}