On the best approximation of some classes of periodic functions in the space L_2

We consider the set L_2^((r)) 2π-periodic functions f∈L_2, whose (r-1)-th order derivative is absolutely continuous, and the r-th order derivative f^((r))∈L_2. We solve the extremal problem of finding an exact Jackson-Stechkin type constant that connects the best polynomial approximation of functions from L_2^((r)) with the average value of the generalized m-th order modulus of continuity of their derivative f^((r)) in the space L_2. We also consider the classes W_(m )^((r)) (u) and W_m^((r)) (u,Φ) of functions from L_2^((r)) such that the average value of the generalized m-th order modulus of continuity of their derivative f^((r)) is bounded from above by unity and, accordingly, by the value of some function Φ(u). We calculate the exact values of the known n-widths (according to Bernstein, to Gelfand, to Kolmogorov, linear, and projection) of the class W_(m )^((r) ) (u). Then we solve the extremal problem of finding the exact value of the best approximation for the class W_m^((r)) (u,Φ). The obtained results develop and complement some known results on the best approximation of various classes of functions in L_2. In the paper, we use methods for solving extremal problems in normed spaces, as well as the method developed by V.M. Tikhomirov for estimating from below the n-widths of functional classes in Banach spaces.