Rate distortion functions of countably infinite alphabet memoryless sources

The Shannon lower bound approach to the evaluation of rate distortion functions R(D) for countably infinite alphabet memoryless sources is considered. Sufficient conditions based on the Contraction Mapping Theorem for the existence of the Shannon lower bound RL(D) to R(D) in a region of distortion [0, D1], D1 > 0 are obtained. Sufficient conditions based on the Schauder Fixed Point Theorem for the existence of a Dc > 0 such that R(D) = RL(D) for all D ε [0, Dc] are derived. Explicit evaluation of R(D) is considered for a class of column balanced distortion measures. Other results for distortion measures with no symmetry conditions are also discussed