Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems

Assume that A(U) is the set of functions analytic in the disk U:={z:vertical bar z vertical bar<1} , L-2((r)) :=L-2((r)) (U) for r is an element of N is the class of functions f is an element of A(U) such that f((r)) is an element of L-2((r)) , and W-(r) L-2 is the class of functions f is an element of L-2((r)) satisfying the constraint parallel to f((r))parallel to <= 1 . We find exact values for mean-square approximations of functions f is an element of W-(r) L-2 and their successive derivatives f((s)) (1 <= s <= r-1 , r >= 2 ) in the metric of the space L-2 . A similar problem is solved for the class W-2((r)) (K-m ,psi) (r is an element of Z(+) , m is an element of N ) of functions f is an element of L-2((r)) such that the K-functional of their r th derivative satisfies the condition K-m (f((r)),t(m)) <= psi(t(m)), 0 < t < 1, where psi is some increasing majorant and psi(0) = 0.