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 Mat. Zametki, 2011, Volume 90, Issue 3, Pages 351–361 (Mi mz8545)

Approximation of Classes of Convolutions by Linear Operators of Special Form

V. P. Zastavnyia, V. V. Savchukb

a Donetsk National University
b Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: A parametric family of operators $G_\rho$ is constructed for the class of convolutions $\mathbf{W}_{p,m}(K)$ whose kernel $K$ was generated by the moment sequence. We obtain a formula for evaluating
$$E(\mathbf{W}_{p,m}(K);G_\rho)_p:=\sup_{f\in\mathbf{W}_{p,m}(K)}\|f-G_\rho(f)\|_p.$$
For the case in which $\mathbf{W}_{p,m}(K)=\mathbf{W}^{r,\beta}_{p,m}$, we obtain an expansion in powers of the parameter $\varepsilon=-\ln\rho$ for $E(\mathbf{W}^{r,\beta}_{p,m};G_{\rho,r})_p$, where $\beta\in\mathbb{Z}$, $r>0$, and $m\in\mathbb{N}$, while $p=1$ or $p=\infty$.

Keywords: convolution, linear operator, periodic measurable function, moment sequence, Borel measure, Fourier series, Euler polynomial, Bernoulli numbers

DOI: https://doi.org/10.4213/mzm8545

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English version:
Mathematical Notes, 2011, 90:3, 333–343

Bibliographic databases:

UDC: 517.518.83+517.15
Revised: 16.03.2011

Citation: V. P. Zastavnyi, V. V. Savchuk, “Approximation of Classes of Convolutions by Linear Operators of Special Form”, Mat. Zametki, 90:3 (2011), 351–361; Math. Notes, 90:3 (2011), 333–343

Citation in format AMSBIB
\Bibitem{ZasSav11} \by V.~P.~Zastavnyi, V.~V.~Savchuk \paper Approximation of Classes of Convolutions by Linear Operators of Special Form \jour Mat. Zametki \yr 2011 \vol 90 \issue 3 \pages 351--361 \mathnet{http://mi.mathnet.ru/mz8545} \crossref{https://doi.org/10.4213/mzm8545} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2868365} \transl \jour Math. Notes \yr 2011 \vol 90 \issue 3 \pages 333--343 \crossref{https://doi.org/10.1134/S0001434611090033} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296476500003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80155149514} 

• http://mi.mathnet.ru/eng/mz8545
• https://doi.org/10.4213/mzm8545
• http://mi.mathnet.ru/eng/mz/v90/i3/p351

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This publication is cited in the following articles:
1. O. L. Vinogradov, “Sharp estimates of best approximations by deviations of Weierstrass-type integrals”, J. Math. Sci. (N. Y.), 194:6 (2013), 628–638
2. Prestin J., Savchuk V.V., Shidlich A.L., “Direct and Inverse Theorems on the Approximation of 2 Pi-Periodic Functions By Taylor-Abel-Poisson Operators”, Ukr. Math. J., 69:5 (2017), 766–781
3. R. M. Trigub, “Asymptotics of approximation of continuous periodic functions by linear means of their Fourier series”, Izv. Math., 84:3 (2020), 608–624
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