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

K. Runovski ; H.-J. Schmeisser

Eurasian mathematical journal, Tome 1 (2010) no. 3, pp. 112-133

Résumé

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.