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Mat. Zametki, 1976, Volume 19, Issue 3, Pages 353–364 (Mi mz7754)  

Direct and inverse inequalities for $\varphi$-Fejér mean-square approximations

V. Yu. Popov

Mathematics and Mechanics Institute of the Ural Scientific Center, Academy of Sciences of the USSR

Abstract: We consider approximation of a function $f\in W_2^l(R_1)$, $l\ge0$, by linear operators of the form
$$ K_\sigma^\varphi(f;x)=\frac1{\sqrt{2\pi}}\int_{R_1}\varphi(\frac u\sigma)\widetilde f(u)e^{iux} du,\quad \sigma>0. $$
We elucidate the conditions for the existence of direct and inverse inequalities between the quantities $\|f-K_\sigma^\varphi(f)\|_{L_2}$ and $\omega_k(f;\tau/\sigma)_{L_2}$, viz., the $k$-th integral modulus of continuity of the function $f(x)$, $k=1,2,…,$. Under some restrictions on $\varphi(u)$, $u\in R_1$ the exact constants in these inequalities are found.

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English version:
Mathematical Notes, 1976, 19:3, 213–220

Bibliographic databases:

UDC: 517.5
Received: 11.06.1974

Citation: V. Yu. Popov, “Direct and inverse inequalities for $\varphi$-Fejér mean-square approximations”, Mat. Zametki, 19:3 (1976), 353–364; Math. Notes, 19:3 (1976), 213–220

Citation in format AMSBIB
\Bibitem{Pop76}
\by V.~Yu.~Popov
\paper Direct and inverse inequalities for $\varphi$-Fej\'er mean-square approximations
\jour Mat. Zametki
\yr 1976
\vol 19
\issue 3
\pages 353--364
\mathnet{http://mi.mathnet.ru/mz7754}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=425477}
\zmath{https://zbmath.org/?q=an:0355.41023|0345.41009}
\transl
\jour Math. Notes
\yr 1976
\vol 19
\issue 3
\pages 213--220
\crossref{https://doi.org/10.1007/BF01437854}


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