Qualitative properties of solutions to \(H^{\infty}\)-optimization problems
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.
scientific article; zbMATH DE number 4060118
scientific article