In this dissertation, we discuss a number of results on superoptimal approximation by analytic and meromorphic matrix-valued functions on the unit circle. We first prove the existence of a monotone non-decreasing thematic factorization for admissible (e.g. continuous) very badly approximable matrix functions. Unlike the case of monotone non-increasing thematic factorizations, it is shown that thematic indices in a monotone non-decreasing thematic factorization are not uniquely determined. We then consider the problem of characterizing superoptimal singular values. An extremal problem is introduced and its connection with the sum of superoptimal singular values is explored by considering a new clans of operators: Hankel-type operators on Hardy spaces of matrix functions. Lastly, we consider approximation by meromorphic matrix-valued functions; the so-called Nehari-Takagi problem. We provide a counterexample that shows that the index formula in connection with meromorphic approximation, which is well-known to hold in the case of scalar-valued functions, fails in the case of matrix-valued functions.
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