A semi-linear space is a linear space lacking the existence of the opposite elements and of the distributive law \((\lambda+\mu)x= \lambda x+\mu x\). An \(L\)-space is a semilinear space \(X\) endowed with a metric \(h_X\) such that (i) \(h_X(\alpha x,\alpha y)=\vert \alpha \vert h_X(x,y)\) and (ii) \(h_X(x+z,y+z)\le h_X(x,y)\) for all \(x,y,z\in X\) and \(\alpha \in\mathbb{R}\). The \(L\)-space \((X,h_X)\) is called isotropic if (ii) holds with equality. An element \(x\in X\) is called convex if \((\alpha+\beta)x= \alpha x+\beta x\) for all \(\alpha,\beta\ge 0\). A typical example is the space \(\Omega(X)\) of all nonempty closed bounded subsets of a normed linear space \(X\) with the Hausdorff metric. In this case, the convex elements are the convex sets in \(\Omega(X)\).
The paper is concerned with the problem of optimal recovery of the operators on the classes \(H^\omega(T,X)\) of continuous functions defined on a metric compact \(T\), taking values in an \(L\)-space \(X\) and having a given majorant \(\omega(t) \) of their modulus of continuity. The operators are recovered based on the values (known with errors) of the functions on a finite set of points.
From the text: “The consideration of operators defined on function \(L\)-spaces allows us to include many important operators acting in the spaces of functions with values in Banach spaces (in particular in the spaces of random processes), spaces of set-valued functions and functions with fuzzy values, and obtain new results on the optimal recovery of such operators.”