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Báo cáo toán học: "Lie groups over the field of rational functions, signed spectral factorization, signed interpolation, and amplifier design "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Lie nhóm trên các lĩnh vực chức năng hợp lý, đã ký thừa quang phổ, nội suy ký kết, và thiết kế bộ khuếch đại. | Copyright by INCREST 1982 J. OPERATOR THEORY 8 1982 19-64 LIE GROUPS OVER THE FIELD OF RATIONAL FUNCTIONS SIGNED SPECTRAL FACTORIZATION SIGNED INTERPOLATION AND AMPLIFIER DESIGN JOSEPH A. BALL and J. WILLIAM HELTON INTRODUCTION This paper concerns a Lie group of In X 2 matrices over the field St of functions on the unit circle with rational continuations to the complex plane. The group we will study consists of all such matrices which satisfy g Ỉ 0 0 I - Jg o 0 -jj and is denoted by SlU n n . We also investigate a particular semigroup Ổ t7 n n consisting of members of 0tU n n which satisfy certain analyticity properties. The study of ăỉU n n is closely bound up with classical Nevanlinna-Pick interpolation theory extended to Grassmannian valued functions. Although the paper is entirely mathematical the motivation for it is physical. The desire was to build the mathematical machinery appropriate for a systematic theory of amplifier design. So the last two sections of the paper concern an optimization problem which physically amounts to the design of a linearized transistor amplifier with maximum gain over all frequencies. While we were not able to fully solve the problem we do make a significant reduction and it is reasonable to believe that some of the main theorems herein will be a part of a unified theory if one ever exists. Now we state our main mathematical results. The group SiU n n acts via the linear fractional map am P xm b y -1 on MMn the n X n matrices with entries from here g e U n n y and m G Mn. The first undertaking in this article is to determine some basics about 20 JOSEPH A. BALL and J. WILLIAM HELTON the orbits of this transformation group. In particular we determine the orbits of the constant elements in S Mn and find that they are just what one would hope. To describe these orbits define k I to be the matrix functions in Mfj k. J whose values at each point on the circle are matrices with j singular values less than 1 k singular values equal to 1