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Convergence of Best Entropy Estimates

Given a finite number of moments of an unknown density on a finite measure space, the best entropy estimate--that nonnegative density x with the given moments which minimizes the Boltzmann- Shannon entropy I(x):= x log x--is considered. A direct proof is given that I has the Kadec property in L--if Yn converges weakly to 37 and I(yn) converges to I(37), then yn converges to 37 in norm. As a corollary, it is obtained that, as the number of given moments increases, the best entropy estimates converge in LI norm to the best entropy estimate of the limiting problem, which is simply in the determined case. Furthermore, for classical moment problems on intervals with strictly positive and sufficiently smooth, error bounds and uniform convergence are actually obtained.