Generalized projections for non-negative functions
Given a strictly convex function \(f\) and a \(\sigma\)-finite measure space \((X, {\mathcal X}, \mu)\), the \(f\)-divergence \(D_f\) and the Bergman distance are used as distances on the set \(S\) of \({\mathcal X}\)-measurable nonnegative functions. An important problem in many applications of such distances in probability theory, statistics and information theory is the question of the existence of projections of elements \(s \in S\) onto convex subsets \(E \subseteq S\). The aim of the paper is to generalize results for \(I\)-projections \((f = x \ln x)\) earlier obtained by the author to the distances \(D_f\) and \(B_f\) for general \(f\). The existence of projections and an inequality being a generalization of Pythagoras' theorem are established in Theorem 1. The convergence in \(\mu\)-measure and in \(L_1 (\mu)\) of minimizing sequences is investigated in Theorem 2. Convex subsets \(E \subseteq S\) defined by a system of linear inequalities are considered in Theorem 3.
scientific article; zbMATH DE number 794425