Topological aspects of the caratheodory problem

This paper presents new results on the problem of realization of a power spectrum of a stochastic process from a given set of error-free covariance samples (spectral estimation). This problem is known in the mathematical literature as the Caratheodory problem and has been studied extensively. We present a novel topological approach (as opposed to algebraic and function-theoretic ones that have been developed classically) which enables us to address certain questions about the degree of the interpolating functions, or, equivalently, the dimension of associated stochastic realizations. A new class of recursive spectral estimators is also introduced. These are tailored to make use of the nonuniqueness that is inherent when obtaining realizations of a certain dimension.