Qualitative properties of solutions to \(H^{\infty}\)-optimization problems
type: Item
instance of: scholarly article
title: Qualitative properties of solutions to \(H^{\infty}\)-optimization problems
zbMATH Open document ID: 0649.46050
DOI: 10.1016/0022-1236(87)90099-1
author: Stefen Hui
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let \(H^ p\) denote the space of functions in \(L^ p(d\theta/2\pi)\), \(1\leq p\leq \infty\), that have vanishing negative Fourier coefficients, and A be the set of continuous functions in \(H^{\infty}\). Suppoe \(\Gamma (e^{i\theta},z)\) is a nonnegative function of \(e^{i\theta}\) and z, where \(\theta\in [0,2\pi]\) and \(z\in {\mathbb{C}}\). Let \(OPT(E)=\inf \{\| \Gamma (e^ i,f(e^{i\theta}))\|_{\infty}:f\in E\}\), where E is either \(H^{\infty}\) or A. The author's main objective is to study the qualitative aspects of functions f for which \(\| \Gamma (e^{i\theta},f(e^{i\theta}))\|_{\infty}=OPT(E)\). Such a function will be called an optimum for OPT(E). A priori it is not known that optimum exist, and it is not required that optimums be in E. The author briefly indicates how the problem he is intested in arises in the design of linear systems. Conditions on \(\Gamma\) that will guarantee existence, uniqueness, and continuity of solutions to the above equation are given. To obtain results, he considers the special case of \(\Gamma(e^{i\theta},z)=\inf \{s>0:g(e^{i\theta})-z\in s\Omega_{\theta}\}\). Here \(g\in L^{\infty}(d\theta/2\pi)\), and for each \(\theta\in [0,2\pi]\), \(\Omega_{\theta}\) is open, convex, and \(\Delta (0,r)\subset \Omega_{\theta}\subset \Delta(0,R)\) for some \(r>0\) and \(R>0\) independent of \(\theta\). Among the results in this paper, one reads that f is continuous if g is Dini-continuous. The paper contains a generalization of this Carleson-Jacobs theorem.
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 49J27
zbMATH DE Number: 4060118
zbMATH Keywords: H\({}^{\infty }\)-optimization problems
zbMATH Keywords: optimum
zbMATH Keywords: Dini-continuous
zbMATH Keywords: Carleson-Jacobs theorem
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90099-1
OpenAlex ID: W1974422702
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description: scientific article; zbMATH DE number 4060118