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Superoptimal approximation by meromorphic functions

Published online by Cambridge University Press:  24 October 2008

V. V. Peller  and
N. J. Young
V. V. Peller
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66502, U.S.A.
N. J. Young
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF
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Abstract

Let G be a matrix function of type m × n and suppose that G is expressible as the sum of an H∞ function and a continuous function on the unit circle. Suppose also that the (k – 1)th singular value of the Hankel operator with symbol G is greater than the kth singular value. Then there is a unique superoptimal approximant to G in : that is, there is a unique matrix function Q having at most k poles in the open unit disc which minimizes s∞(G – Q) or, in other words, which minimizes the sequence

with respect to the lexicographic ordering, where

and Sj(·) denotes the jth singular value of a matrix. This result is due to the present authors [PY1] in the case k = 0 (when the hypothesis on the Hankel singular values is vacuous) and to S. Treil[T2] in general. In this paper we give a proof of uniqueness by a diagonalization argument, a high level algorithm for the computation of the superoptimal approximant and a recursive parametrization of the set of all optimal solutions of a matrix Nehari—Takagi problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 3 , April 1996 , pp. 497 - 511
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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