A variant of Stechkin's problem on the best approximation of a fractional order differentiation operator on the axis

A solution is given to Stechkin's problem on the best approximation on the real axis of differentiation operators of fractional (more precisely, real) order $k$ in the space $L_2$ by bounded linear operators from the space $L$ to the space $L_2$ on the class of functions whose fractional derivative of order $n$, $0\le k<n,$ is bounded in the space $L_2$. An upper estimate is obtained for the best constant in the corresponding Kolmogorov inequality. It is shown that the well-known Stechkin lower estimate for the value of the problem of approximating the differentiation operator via the best constant in the Kolmogorov inequality is strict in this case; in other words, Stechkin's problem and the Kolmogorov inequality are not consistent.