Sharp estimates of approximation of periodic functions in Hölder spaces

•Sharp estimates of approximation of periodic functions in Hölder spaces.•Improvements of the direct and inverse theorems of approximation in Hölder spaces.•New estimates of the error of the best approximation in Hölder spaces.•The criteria of precise order of decrease of the best approximation in Hölder spaces.•Strong converse inequalities for general methods of approximation in Hölder spaces.

The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the Hölder spaces Hpr,α for all 0