Sharp Inequalities between the Best Root-Mean-Square Approximations of Analytic Functions in the Disk and Some Smoothness Characteristics in the Bergman Space

In Jackson-Stechkin type inequalities for the smoothness characteristic Lambda(m) (f), m is an element of N, we find exact constants determined by averaging the norms of finite differences of mth order of a function f is an element of B-2. We solve the problem of best joint approximation for a certain class of functions from B-2((r)), r is an element of Z(+) whose smoothness characteristic Lambda(m) (f) averaged with a given weight is bounded above by the majorant Phi. The exact values of n-widths of some classes of functions are also calculated.