Optimal Recovery of Operators in Function L-Spaces

Abstract

We consider the classes of functions that are defined on a metric compact, take values in an L-space (i.e. semi-isotropic semi-linear metric space) and have a given majorant of modulus of continuity. For a wide class of operators Λ that act on such function classes, we solve the problem of the optimal recovery based on inaccurate values of the functions in a finite number of points. As an application of the obtained results, we give the solutions of the problems of optimal recovery of the solutions for operator equation of the form x = f + Λx, and in particular for the Fredholm, Volterra, and Volterra-Fredholm integral equations of the second kind for functions with values in L-spaces. As consequences of the results in the article, one can obtain new results for optimal recovery of the operators acting in the spaces of set-valued and fuzzy-valued functions, as well as in the spaces of functions with values in Banach spaces; in particular, in the spaces of random processes.