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This article is cited in 1 scientific paper (total in 1 paper)
Approximation of continuous functions by trigonometric polynomials almost everywhere
T. V. Radoslavova V. A. Steklov Mathematical Institute, USSR Academy of Sciences
Abstract:
We consider the problem of the rate of approximation of continuous $2\pi$-periodic functions of class $W^rH[\omega]_C$ by trigonometric polynomials of order $n$ on sets of total measure. We prove that when $r\ge0$, $\omega(\delta)\delta^{-1}\to\infty$ ($\delta\to0$) there exists a function $f\in W^rH[\omega]_C$ such that $\widetilde f\in W^rH[\omega]_C$ and for any sequence $\{t_n\}_{n=1}^\infty$ we have almost everywhere on $[0,2\pi]$
\begin{gather*}
\varlimsup_{n\to\infty}|f(x)-t_n(x)|n^r\omega^{-1}(1/n)>C_x>0
\\
\varlimsup_{n\to\infty}|\widetilde f(x)-t_n(x)|n^r\omega^{-1}(1/n)>C_x>0
\end{gather*}
Received: 24.09.1975
Citation:
T. V. Radoslavova, “Approximation of continuous functions by trigonometric polynomials almost everywhere”, Mat. Zametki, 19:1 (1976), 49–62; Math. Notes, 19:1 (1976), 29–36
Linking options:
https://www.mathnet.ru/eng/mzm7722 https://www.mathnet.ru/eng/mzm/v19/i1/p49
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