Korneichuk-Stechkin Lemma, Ostrowski and Landau inequalities, and optimal recovery problems for $L$-space Valued Functions

TL;DR

This paper extends classical inequalities and recovery problems to functions valued in L-spaces, providing sharp bounds and solutions for optimal recovery of functions, derivatives, and operators, with applications to fuzzy and Banach space-valued functions.

Contribution

It introduces an analogue of the Korneichuk--Stechkin lemma for L-space valued functions and solves various extremal recovery problems, unifying approaches for multiple function classes.

Findings

Established sharp Ostrowski type inequalities for L-space valued functions.

Solved optimal recovery problems for functions, derivatives, and operators.

Derived Landau type inequalities and addressed Stechkin's approximation problem.

Abstract

We prove an analogue of the Korneichuk--Stechkin lemma for functions with values in $L$-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as well as the problem of integral recovery, on the classes of $L$-space valued functions with given majorant of modulus of continuity. The recovery is done based on $n$ mean values of the functions over some intervals. Moreover, on the classes of functions with given majorant of modulus of continuity of their Hukuhara type derivative, we solve the problem of optimal recovery of the function and the Hukuhara type derivative. The recovery is done based on $n$ values of the function. We also obtain some sharp Landau type inequalities and solve an analogue of the Stechkin problem about approximation of unbounded operators by bounded ones and the problem of optimal recovery of an unbounded operator on a class of elements, known with error. Consideration of $L$-space valued functions gives a unified approach to solution of the mentioned above extremal problems for the classes of multi- and fuzzy-valued functions as well as for the classes of functions with values in Banach spaces, in particular random processes, and many other classes of functions.