CONTINUITY PROPERTIES OF

BEST ANALYTIC APPROXIMATION

Vladimir Peller Nicholas J. Young

Abstract. Let A be the operator which assigns to each m × n matrix-valued function on the unit circle with entries in H^{∞} + C its unique superoptimal approximant in the space of bounded analytic m×n matrix-valued functions in the open unit disc. We study the continuity of A with respect to various norms. Our main result is that, for a class of norms satifying certain natural axioms, A is continuous at any function whose superoptimal singular values are non-zero and is such that certain associated integer indices are equal to 1. We also obtain necessary conditions for continuity of A at point and a sufficient condition for the continuity of superoptimal singular values.

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