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 Eurasian Math. J., 2010, Volume 1, Number 3, Pages 112–133 (Mi emj31)

On convergence of families of linear polynomial operators generated by matrices of multipliers

K. Runovskia, H.-J. Schmeisserb

a Sevastopol Branch of Moscow State University, Sevastopol, Ukraine
b Mathematisches Institut Friedrich-Schiller University, Jena, Germany

Abstract: The convergence of families of linear polynomial operators with kernels generated by matrices of multipliers is studied in the scale of the $L_p$-spaces with $0<p\le+\infty$. An element $a_{n, k}$ of generating matrix is represented as a sum of the value of the generator $\varphi(k/n)$ and a certain “small” remainder $r_{n, k}$. It is shown that under some conditions with respect to the remainder the convergence depends only on the properties of the Fourier transform of the generator $\varphi$. The results enable us to find explicit ranges for convergence of approximation methods generated by some classical kernels.

Keywords and phrases: trigonometric approximation, convergence, Fourier multipliers, Jackson, Cesaro and Fejér–Korovkin kernels.

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MSC: 42A10, 42A45, 42B99
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Citation: K. Runovski, H.-J. Schmeisser, “On convergence of families of linear polynomial operators generated by matrices of multipliers”, Eurasian Math. J., 1:3 (2010), 112–133

Citation in format AMSBIB
\Bibitem{RunSch10} \by K.~Runovski, H.-J.~Schmeisser \paper On convergence of families of linear polynomial operators generated by matrices of multipliers \jour Eurasian Math. J. \yr 2010 \vol 1 \issue 3 \pages 112--133 \mathnet{http://mi.mathnet.ru/emj31} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2904769} \zmath{https://zbmath.org/?q=an:1217.42003} 

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• http://mi.mathnet.ru/eng/emj/v1/i3/p112

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This publication is cited in the following articles:
1. K. Runovski, H.-J. Schmeisser, “Methods of trigonometric approximation and generalized smoothness. I”, Eurasian Math. J., 2:3 (2011), 98–124
2. K. V. Runovski, “Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness”, Sb. Math., 208:2 (2017), 237–254
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