Korneichuk-Stechkin Lemma, Ostrowski and Landau inequalities, and optimal recovery problems for $L$-space Valued Functions
Vladyslav Babenko, Vira Babenko, Oleg Kovalenko
TL;DR
This work generalizes the Korneichuk--Stechkin framework to $L$-space valued functions, enabling sharp Ostrowski-type inequalities and a suite of optimal recovery results for identities, convexifying operators, and integrals on classes with prescribed modulus of continuity. The authors introduce a robust $L$-space toolkit, including a generalized Korneichuk--Stechkin lemma via $\Sigma$-rearrangements, and apply it to recoveries and inequalities for Hukuhara derivatives, culminating in Landau-type and Stechkin-type results. Key contributions include sharp bounds for $L$-space valued functionals, optimal recovery strategies with mean-value information, and explicit extremals that attain bounds, plus a unifying perspective across Banach-, multivalued-, and fuzzy-valued contexts. The results have broad implications for approximation theory, optimal algorithms, and applications to random processes and fuzzy optimization by providing a common, rigorous framework for extremal problems in $L$-spaces.
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.