Some problems of approximation of periodic functions by trigonometric polynomials in $L_2$

The paper is consists from two parts. In first part summarizes the review of findings on best approximation of periodic functions by trigonometric polynomials in Hilbert space $L_{2}:=L_{2}[0,2\pi]$. The sharp inequalities between the best approximation and averaged with given weights modulus of continuity of $m$th order values $r$th derivatives of functions and analogues for some modified modulus of continuity presented.
In second part, some new sharp Jackson-Stechkin type inequalities for characteristics of smoothness studied by K. V. Runovski and more detail by S. B. Vakarchuk and V. I. Zabutnaya are proposed. The sharp result on joint approximation of function and successive derivatives for some classes of functions defined by modulus of smoothness obtained.