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

For any $t\in [0,1]$, we obtain the exact value of the modulus of continuity
$$ \omega_N(D_t,\delta):=\sup\{|x'(t)|:\|x\|_{L_{\infty}[0,1]}\le \delta,\, \|x''\|_{L_{N}^*[0,1]}\le 1\}, $$
where $L_N^*$ is the dual Orlicz space with Luxemburg norm and $D_t$ 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 $D_t$ by bounded linear operators for the class of functions $x$ such that $\|x''\|_{L_{N}^*[0,1]}\le 1$.