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Nikol'skii known theorem for the kernels satisfying a condition $A^*_n$, isproved and for kernels from wider class. Explicit formulas for calculating thevalue of an approximation of classes $\W^{r, \beta}_{p, n} $ by convolutionoperators of special form are obtained. Here $ \beta\in\Z $, $r> 0 $, $n\in\N$, and $p=1 $ or $p =\infty $. As particular cases obtained explicit formulasfor value of an approximation of the indicated classes generalized Abel-Poissonmeans, biharmonic operators of Poisson, Cesaro and Riesz means. In some casesfor value of an approximation of the indicated classes asymptotic expansions onparameter are found. In case of natural $r$ some results have been obtained inworks Nikol'skii, Nagy, Timan, Telyakovskii, Baskakov, Falaleev, Kharkevich andother 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.

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