Approximation of the differentiation operator by bounded linear operators in the space $L_2$ on the half-line

We will discuss Stechkin's problem on the best approximation of the differentiation operator of order $k$ on the class of $n$ times differentiable functions $(0\le k < n)$ by bounded linear operators in the spaces $L_p$ on the real line and on the half-line as well as the related problem on the optimal recovery of smooth functions given with a known error. A solution of Stechkin's problem will be given in the space $L_2(0,\infty)$ for $k = 1$ and $n = 2;$ this result was obtained jointly with M.A.Filatova.