Entropy Minimization, Hilbert′s Projective Metric, and Scaling Integral Kernels

Suppose that (S, μ) and (T, ν) are given measure spaces with μ(S) < ∞ and ν(T) < ∞. If k ∈ L∞(S × T) is a nonnegative function and α ∈ L1(S) and β ∈ L1(T) are positive almost everywhere, the so-called DAD problem (k, α, β) asks whether there exist ƒ ∈ L1(S) and g ∈ L1(T), f and g positive almost everywhere, with ∫T ƒ(s) k(s, t) g(t) ν(dt) = α(s), μ almost everywhere, and∫S ƒ(s) k(s, t) g(t) μ(ds) = β(t), v almost everywhere. Such a pair (ƒ, g), if it exists, is called a solution of the DAD problem (k, α, β). We present here essentially sharp conditions under which the DAD problem (k, α, β) has a solution. We also give results concerning the uniqueness (to within positive scalar multiples) of solutions (ƒ, g), iterative schemes for approximating solutions, and continuous dependence of solutions on (kα, kβ). Methods of proof involve a mixture of variational methods (entropy minimization) and fixed point theory; Hilbert′s projective metric also plays a useful role. As corollaries of our results we obtain generalizations of a variety of earlier DAD theorems. We are also able to discuss limiting behaviour of sequences of matrix DAD problems, where the dimensions of the matrices approach infinity.