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A detailed study is made of positive definite half-plane Toeplitz systems and of the corresponding least-squares inverse approximaion problems. The general question is to minimize a given functional over the space of two-variable functions with a prescribed half-plane support. For some particular supports, such as those considered by Marzetta, the minimizing functions enjoy remarkable recurrence, stability, and convergence properties. Simple derivations of these properties are given and various new results are obtained. As an application, it is shown how the half-plane spectral factor of a given magnitude function can be inversely approximated by stable pseudepolynomials.