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

Netuka, Ivan

J. Approximation Theory 51, 175-182 (1987).

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

MSC:

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

Keywords:

Dirichlet problem; best uniform approximation

References:

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