Sharp estimates of approximation of periodic functions in Hölder spaces

The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the Hölder spaces H_p^r,α for all 0 < p≤ ∞ and 0 < α ≤ 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,α.