Pervasive function spaces and the best harmonic approximation (Q1100660)

Let \(U\subset {\mathbb{R}}^ m \)be a bounded open set with boundary \(\partial U\). Let H(\(\partial U)\) denote the set of all functions on \(\partial U\) for which there is a solution of the classical Dirichlet problem. Thus \(f\in H(\partial U)\) provided that f has a continuous extension to the closure \(\bar U\) of U which is harmonic in U. It is known that H(\(\partial U)\) is a uniformly closed subspace of the Banach space C(\(\partial U)\) of all continuous functions on \(\partial U\). In general, however H(\(\partial U)\neq C(\partial U)\). The purpose of this paper is to study the following question: given \(f\in C(\partial U)\) can one find from the functions of H(\(\partial U)\) a best uniform approximation to f?