SUPEROPTIMAL ANALYTIC APPROXIMATIONS OF MATRIX FUNCTIONS

We study the approximation of a bounded matrix-valued function G on th e unit circle by functions Q bounded and analytic in the unit disc. We show that if G is continuous then there is a unique Q for which the e rror G - Q has a strong minimality property involving not only the L(i nfinity)-norm of G-Q but also the suprema of its subsequent singular v alues. We obtain structural properties of the error G - Q and show tha t certain smoothness properties of G are inherited by Q (e.g., members hip of Besov spaces). (C) 1994 Academic Press, Inc.