  The main purpose of the paper is to study sharp estimates of approximation of
periodic functions in the H\"older spaces $H_p^{r,\alpha}$ for all
$0<p\le\infty$ and $0<\alpha\le r$. By using modifications of the classical
moduli of smoothness, we give improvements of the direct and inverse theorems
of approximation and prove the criteria for the precise order of decrease of
the best approximation in these spaces. Moreover, we obtained strong converse
inequalities for general methods of approximation of periodic functions in
$H_p^{r,\alpha}$.

Classical Analysis and ODEs