The Kolmogorov and Stechkin Problems for Classes of Functions Whose Second Derivative Belongs to the Orlicz Space

Abstract—For any t ∈ [0, 1], we obtain the exact value of the modulus of continuity ωN(Dt, δ): = sup{|x′(t) | : ‖x‖L∞[0,1] ≤ δ, ‖x′′‖L∗N [0,1] ≤ 1}, where L∗N is the dual Orlicz space with Luxemburg norm and Dt is the operator of differentition at the point t. As an application, we state necessary and sufficient conditions in the Kolmogorov problem for three numbers. Also we solve the Stechkin problem, i.e., the problem of approximating an unbounded operator of differentition Dt by bounded linear operators for the class of functions x such that ‖x′′‖L∗N [0,1] ≤ 1. DOI: 10.1134/S000143461201018