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

V. V. Arestov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 4, pp. 37-54

Résumé

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 $n$, $0\le k$ 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.

Détail

Keywords: fractional order differentiation operator, Stechkin's problem, Kolmogorov inequality, Carlson inequality.

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