tailieunhanh - Báo cáo toán học: "The Nevanlinna-Pick problem for matrix-valued functions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Vấn đề Nevanlinna-Chọn cho ma trận có giá trị chức năng. | J. OPERATOR THEORY 15 1986 239-265 Copyright by INCREST 1986 THE NEVANL1NNA-PĨCK PROBLEM FOR MATRIX-VALUED FUNCTIONS N. J. YOUNG 1. INTRODUCTION Tn recent years some complex interpolation problems with a long history have come once again to the fore because of a remarkable diversity of applications in systems engineering. One version of the Nevanlinna-Pick problem is to minimise the supremum norm over the set of bounded analytic functions in the open unit disc u subject to a finite set of interpolation conditions see 21 . There are applications of the solution of this problem in optimal circuit design going back over 40 years now see 9 but the recent heightening of interest was brought about by results of V. M. Adamyan D. z. Arov and M. G. Krein 1 2 3 on a mathematically equivalent problem formulated in terms of infinite Hankel matrices. Engineers have found uses for these results in the problems of identification and realization 8 and in model reduction and digital filter design 9 10 . The Nevanlinna-Pick problem plays an important role in J. w. Helton s far-reaching application of non-Euclidean functional analysis to electronics 14 15 Evans and Helton have even encountered the problem in modelling fluid retention in the lungs 12 In consequence both of these developments and of progress in operator theory there have been many papers on Nevanlinna-Pick interpolation recently in both engineering and mathematics journals. There are now several alternative mathematical approaches which give a neat and unified treatment of a wide range of interpolation and approximation problems. A powerful approach is based on the ideas of commutant lifting 19 and contractive intertwining dilations 5 which can be traced back to pioneering work of D. Sarason 20 . A very elegant method 6 7 is based on the theory of spaces with indefinite inner product while a more function-theoretic approach using Hankel operators stems from Adamyan et al 1 2 3 All of these allow the extension of the .