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 $$n$$ , $$0\leq k<n$$ , 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.