Superoptimal analytic approximation and exterior products

Abstract

This dissertation concerns the classical problem of finding a bounded analytic function on the unit disc D which approximates a given essentially bounded function G on the unit circle T as well as possible in the L1 norm. In the case that G is a continuous mxn matrix-valued function on T; there is a typically large set of optimal bounded analytic approximants in the L1 norm, and it is therefore natural to study bounded analytic approximants Q on D for which G - Q is minimised in a strengthened sense. One de nes, for j 0; s1 j (G Q) = ess sup z2T sj(G(z) Q(z)); where s0; s1; : : : ; sj are the singular values of a matrix. One then says that a bounded analytic matrix function Q is a superoptimal analytic approximant of G if Q lexicographically minimises the sequence (s1 0 (G Q); s1 1 (G Q); : : : ; ) over all bounded analytic matrix functions. It is known that every continuous matrix-valued function on T has a unique superoptimal analytic approximant AG; moreover, for rational G; there are numerical procedures for the calculation of AG: Existing algorithms are computationally intensive. This thesis introduces a new operator-theoretic technique, based on exterior powers of Hilbert spaces and operators, for the calculation of the superoptimal analytic approximants. The result is a new algorithm which avoids some of the lengthier and potentially more illconditioned steps in previously described algorithms. In particular, the present algorithm does not require the spectral factorisation of matrix-valued positive functions on T.