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This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem à la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler pseudo-distance, which gives rise to a convex optimization problem. After developing the variational analysis, we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem, based on the iteration of a nonlinear map in a bounded subset of the dual space. Our main result is the proof of local convergence of the latter, established as a consequence of the central manifold theorem. Supported by numerical evidence, we conjecture that, in the mentioned bounded set, the convergence is actually global.