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

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 ≤ +∞. An element a_n,k of generating matrix is represented as a sum of the value of the generator ϕ(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 ϕ. The results enable us to find explicit ranges for convergence of approximation methods generated by some classical kernels.