Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Superoptimal approximation by meromorphic functions

V. V. Pellera1 and N. J. Younga2

a1 Department of Mathematics, Kansas State University, Manhattan, Kansas 66502, U.S.A.

a2 Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF

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 S0305004100074375_inline1: that is, there is a unique matrix function Q having at most k poles in the open unit disc which minimizes s(GQ) or, in other words, which minimizes the sequence

S0305004100074375_eqnU001

with respect to the lexicographic ordering, where

S0305004100074375_eqnU002

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.

(Received July 11 1994)

(Revised September 26 1994)

Footnotes

† V. V. Peller'research was supported by an NSF grant in modern analysis