Asymptotically optimal estimates of the \(n\)-widths of bounded analytic functions (Q1342518)

In this paper, optimal lower and upper bounds fort the \(n\)-width of bounded analytic functions are given. The \(n\)-width of a subset \(A\) of a Banach space \(X\) is defined by \[ d_ n(A,X) : = \inf_{X_ n} \sup_{x \in A} \inf_{y \in X_ n} | x - y |, \] where \(X_ n\) runs over all \(n\)-dimensional subspaces of \(X\). Let \(\Delta\) be the unit disk in the complex plane \(\mathbb{C}\), and let \(H^ \infty\) be the Hardy space consisting of all bounded analytic functions on \(\Delta\). With \(A\) we denote the unit ball in \(H^ \infty\). Assume that \(E\) is an arbitrary, simply connected subset of \(\Delta\), and that the boundary \(\delta E\) belongs to \(C^{1, \alpha}\) \((0 < \alpha \leq 1)\). Then, we have \[ \exp \left( - {n \over \text{cap} (E, \Delta)} \right) \leq d_ n \bigl( A,C (E) \bigr) \leq \exp \left( - {n \over \text{cap} (E, \Delta)} \right) \quad \biggl( 1 + O \bigl( \ln (n)/n^{\alpha} \bigr) \biggr) \] for all \(n \in \mathbb{N}\), where \(\text{cap} (E, \Delta)\) is the Green capacity of \(E\) with respect to \(\Delta\). The lower bound follows straightforwardly from \textit{H. Widom} [J. Approximation Theory 5, 343-361 (1972; Zbl 0234.30023)] and \textit{S. D. Fisher} and \textit{Ch. A. Micchelli} [Duke Math. J. 47, 789-801 (1980; Zbl 0451.30023)]. The upper bound is proved using some techniques (introduced in Fisher, Micchelli (1980)) from the theory of Faber polynomials to the setting of Blaschke products in the unit disk.