tailieunhanh - Báo cáo toán học: "Optimization over spaces of analytic functions and the corona problem "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tối ưu hóa trên các không gian chức năng phân tích và vấn đề corona. | J. OPERATOR THEORY 15 1986 359-375 Copyright by INCREST 1986 OPTIMIZATION OVER SPACES OF ANALYTIC FUNCTIONS AND THE CORONA PROBLEM J. WILLIAM HELTON 1. INTRODUCTION A classical pursuit in analysis is the problem of finding the distance of a function g on the circle T to H and characterizing properties of the optima. The problem was solved when distance is in the L2 sense by Szego while in the L case much is known and there are over a hundred articles beginning with Carathéodory--Fejer and Pick in the early 1900 s. In particular when g is continuous there is a direct characterization of the L closest point f0 to g provided f0 is continuous. Namely i lể Lol is constant ii the winding number of g yo about zero is negative. The article has two objectives. The first is to generalize this to nonlinear optimization problems. Indeed we find a strict generalization of this property. It applies to highly nonlinear optimization problems and gives a practical test to determine if a particular continuous function f0 is or is not an optimum. We believe the result can be applied to many engineering situations see for example 10 or 9 . The second objective of the article is to generalize the classical Corona theorem. Indeed this is forced upon US by our study of optimization. The optimization problem this paper analyzes concerns a subset E of íí t C v the continuous CN-valued functions on the unit circle. We let r è H be a function on T X CN and study the optimization problem Find OPT To inf sup - II T . X. e ei eT Here II lloo denotes the usual supremum norm. This article concerns qualitative properties of the optimum. Most of our attention focuses on E 91 the algebra of all functions in C with analytic continuation onto D the unit disk. As we shall see two properties characterize solutions f0 of OPT over E 91. The first property is that the function o flattens r that is rie ofe 0 is constant . in 0. The complete characterization of the optimum for smooth r is. 360 J. WILLIAM