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 \(k\) $𝑘$ in the space \(L_{2}\) $\mathrm{subscript}(𝐿, 2)$ by bounded linear operators from the space \(L\) $𝐿$ to the space \(L_{2}\) $\mathrm{subscript}(𝐿, 2)$ on the class of functions whose fractional derivative of order  \(n\) $𝑛$ , \(0\leq k<n\) $(0\le 𝑘)\land (< 𝑛)$ , is bounded in the space  \(L_{2}\) $\mathrm{subscript}(𝐿, 2)$ . 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.