On the fixed point problem in the abstract rate distortion theory

Abstract

This paper introduces a general framework for dealing with the subject of rate distortion or source coding with fidelity criterion, in Polish spaces. The problem is ad-dressed using a variational approach on the space of measure valued functions. Due to the implicit form of the optimal solution, the question of existence of solution is crucial. This is formulated as a fixed point problem on a suitable locally convex topological vector space (consisting of functions with values in the space of finitely additive measures). Existence of solution of the original problem is proved by establishing the existence of a fixed point.