Exact estimation of an approximation of some classes of differentiable functions by convolution operators

Nikol’skii known theorem for the kernels satisfying a condition $A_n^{\ast }$, is proved and for kernels from wider class. Explicit formulas for calculating the value of an approximation of classes ${\text{W}}_{p,n}^{r,\beta }$ by convolution operators of special form are obtained. Here $\beta \in \mathbb{Z}$, $r>0$, $n\in \mathbb{N}$, and $p=1$ or $p=\mathrm{\infty }$. As particular cases obtained explicit formulas for value of an approximation of the indicated classes generalized Abel-Poisson means, biharmonic operators of Poisson, Cesaro and Riesz means. In some cases for value of an approximation of the indicated classes asymptotic expansions on parameter are found. In case of natural $r$ some results have been obtained in works Nikol’skii, Nagy, Timan, Telyakovskii, Baskakov, Falaleev, Kharkevich and other mathematicians. Key words: Nikol’skii theorem, an approximation of classes of functions, Abel-Poisson means, biharmonic operators of Poisson, Riesz and Cesaro means, asymptotic expansion, multiply monotone function, Hurwitz function.