Pervasive function spaces and the best harmonic approximation. (English) Zbl 0641.41025

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?
Reviewer: H.R.Dowson


41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A30 Approximation by other special function classes
41A50 Best approximation, Chebyshev systems
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