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Continuity properties of best analytic approximation. (English) Zbl 0890.47016

Beginning with the fundamental work of V. M. Adamyan, D. Z. Arov and M. G. Krein [see for instance their paper “Infinite Hankel block matrices and some related continuation problems”, Izv. Akad. Nauk Armyan S.S.R. Ser. Mat. 6, 87-112 (1971; Zbl 0311.15012)] the interplay between the theory of Hankel operators, distance formulas in function theory in the unit disk and some problems of optimal control became a major research area by itself. The paper under review is a remarkable contribution in this direction.
Without entering into complicated details, the main problem is the continuity of the best superoptimal approximant (in the space of rectangular matrices with entries bounded analytic functions in the unit disk) of similar matrices with entries in the algebra \(H^\infty+C\) (that is bounded analytic functions plus continuous functions). Already the scalar case corresponds to a classical interpolation problem in function theory. The authors introduce a class of natural norms with respect to which the continuity of the approximant holds under simple conditions involving the singular values (depending on a parameter) or other similar invariants.
The paper is a synthesis of prior classical work and recent advances in the mathematical theory of linear systems. For some related results, see V. V. Peller and N. J. Young, J. Funct. Anal. 120, No. 2, 300-343 (1994; Zbl 0808.47011)].

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
41A35 Approximation by operators (in particular, by integral operators)
30E10 Approximation in the complex plane
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
47A57 Linear operator methods in interpolation, moment and extension problems
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