Geodesic
On convergence of families of linear polynomial operators generated by matrices of multipliers
Eurasian mathematical journal, Tome 1 (2010) no. 3, pp. 112-133.

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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 $0$. 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. 
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K. Runovski; H.-J. Schmeisser. On convergence of families of linear polynomial operators generated by matrices of multipliers. Eurasian mathematical journal, Tome 1 (2010) no. 3, pp. 112-133. http://geodesic.mathdoc.fr/item/EMJ_2010_1_3_a6/

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