  Nikol'skii known theorem for the kernels satisfying a condition $A^*_n$, is
proved and for kernels from wider class. Explicit formulas for calculating the
value of an approximation of classes $\W^{r, \beta}_{p, n} $ by convolution
operators 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 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.
