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Matematicheskie Zametki, 1975, Volume 18, Issue 4, Pages 527–539 (Mi mzm9967)  

Local properties of functions and approximation by trigonometric polynomials

T. V. Radoslavova

V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR
Abstract: Suppose $\Phi_{p,E}$ ($p>0$ an integer, $E\subset[0,2\pi]$) is a family of positive nondecreasing functions $\varphi_x(t)$ ($t>0$, $x\in E$) such that $\varphi_x(nt)\leqslant n^p\varphi_x(t)$ ($n=0,1,\dots$), $t_n$ is a trigonometric polynomial of order at most $n$, and $\Delta_h^l(f,x)$ ($l>0$ an integer) is the finite difference of order $l$ with step $h$ of the function $f$. \underline{THEOREM.} Suppose $f(x)$ is a function which is measurable, finite almost everywhere on $[0, 2\pi]$, and integrable in some neighborhood of each point $x\in E$, $\varphi_x\in\Phi_{p,E}$ and
$$ \varlimsup_{\delta\to\infty}\left|(2\delta)^{-1}\int_{-\delta}^\delta\Delta_u^l(f,x)\,du\right|\varphi_x^{-1}(\delta)\leqslant C(x)<\infty\qquad(x\in E). $$
Then there exists a sequence $\{t_n\}_{n=1}^\infty$, which converges to $f(x)$ almost everywhere, such that for $x\in E$
$$ \varlimsup_{n\to\infty}|f(x)-t_n(x)|\varphi_x^{-1}(1/n)\leqslant AC(x), $$
where $A$ depends on $p$ and $l$.
Received: 18.06.1975
English version:
Mathematical Notes, 1975, Volume 18, Issue 4, Pages 903–910
DOI: https://doi.org/10.1007/BF01153042
Bibliographic databases:
Document Type: Article
UDC: 517.5
Language: Russian
Citation: T. V. Radoslavova, “Local properties of functions and approximation by trigonometric polynomials”, Mat. Zametki, 18:4 (1975), 527–539; Math. Notes, 18:4 (1975), 903–910
Citation in format AMSBIB
\Bibitem{Rad75}
\by T.~V.~Radoslavova
\paper Local properties of functions and approximation by trigonometric polynomials
\jour Mat. Zametki
\yr 1975
\vol 18
\issue 4
\pages 527--539
\mathnet{http://mi.mathnet.ru/mzm9967}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=393998}
\zmath{https://zbmath.org/?q=an:0351.42003}
\transl
\jour Math. Notes
\yr 1975
\vol 18
\issue 4
\pages 903--910
\crossref{https://doi.org/10.1007/BF01153042}
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