Generalized Smoothness Characteristics in Jackson-Type Inequalities and Widths of Classes of Functions in $L_2$

In the space $L_2$, we study a number of smoothness characteristics of functions; our study is based on the use of the generalized shift operator $\tau_h$. For the case in which $\tau$ is the Steklov operator $S$, we obtain exact constants in Jackson-type inequalities for some classes of $2\pi$-periodic functions. We also calculate the exact values of the $n$-widths of function classes defined by the smoothness characteristics under consideration.