Kolmogorov type inequalities for the Marchaud fractional derivatives on the real line and the half-line

TL;DR

This paper develops new inequalities for Marchaud and Hadamard fractional derivatives on the real line and semi-line, addressing approximation and recovery of unbounded operators within these frameworks.

Contribution

It introduces novel Kolmogorov type inequalities for fractional derivatives and solves related approximation and recovery problems for unbounded operators.

Findings

Established new inequalities for fractional derivatives.

Solved the Stechkin approximation problem for unbounded operators.

Provided methods for optimal operator recovery with prescribed error.

Abstract

In this paper we establish some new Kolmogorov type inequalities for the Marchaud and Hadamard fractional derivatives of functions defined on a real axis or semi-axis. Simultaneously we solve two related problems: the Stechkin problem on the best approximation of unbounded operators by bounded ones on a given class of elements and the problem of optimal recovery of operator on elements from some class given with prescribed error. Keywords: inequalities for derivatives, fractional derivatives, approx- imation of unbounded operators by bounded ones, optimal recovery of operators, ideal lattice.