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

 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

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}