Sharp inequalities of Jackson-Stechkin type for periodic functions in $L_2$ differentiable in the Weyl sense

For periodic functions differentiable in the sense of Weyl and belonging to the space $L_{2}$, sharp inequalities of Jackson–Stechkin type are obtained for a special $m$th-order modulus of continuity generated by the Steklov operator (function). Similar characteristics of smoothness of functions were considered earlier by V. A. Abilov, F. V. Abilova, V. M. Kokilashvili, S. B. Vakarchuk, V. I. Zabutnaya, K. Tukhliev, etc. For classes of functions defined in terms of these characteristics, we solve a number of extremal problems of polynomial approximation theory.