# zbMATH — the first resource for mathematics

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

##### 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
Full Text:
##### References:
 [1] Boboc, N; Mustata, P, Espaces harmoniques associés aux operateurs différentiels linéaires du second ordre de type elliptiques, () · Zbl 0167.40301 [2] Brelot, M, Éléments de la théorie classique du potentiel, (1961), CDU Paris · Zbl 0084.30903 [3] Burchard, H.G, Best uniform harmonic approximation, approximation theory II, (), 309-314 · Zbl 0363.41029 [4] Constantinescu, C; Cornea, A, Potential theory on harmonic spaces, (1972), Springer-Verlag Berlin · Zbl 0248.31011 [5] Čerych, J, A word on pervasive function spaces, (), 107-109, Sofia [6] Effros, E; Kazdan, J.L, Applications of Choquet simplexes to elliptic and parabolic boundary value problems, J. differential equations, 8, 95-134, (1970) · Zbl 0255.46018 [7] Goldstein, M; Haussmann, W; Jetter, K, Best harmonic L1 approximation to sub-harmonic functions, J. London math. soc., 30, 257-264, (1984) · Zbl 0705.31002 [8] Hayman, W.K; Kershaw, D; Lyons, T.J, The best harmonic approximant to a continuous function, (), 317-327 [9] Helms, L.L, Introduction to potential theory, (1969), Wiley-Interscience New York · Zbl 0188.17203 [10] Hervé, R.-M, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. inst. Fourier (Grenoble), 12, 415-571, (1962) · Zbl 0101.08103 [11] Köhn, J; Sieveking, M, Reguläre and extremale randpunkte in der potentialtheorie, Rev. roumaine math. pures appl., 12, 1489-1502, (1967) · Zbl 0158.12804 [12] de la Pradelle, A, Approximation et caractère de quasi-analycité dans la théorie axiomatique des fonctions harmoniques, Ann. inst. Fourier (Grenoble), 17, 383-399, (1967) · Zbl 0153.15501 [13] Rudin, W, Real and complex analysis, (1974), McGraw Hill New York [14] Singer, I, Best approximation in normed linear spaces by elements of linear subspaces, (1970), Springer-Verlag Berlin · Zbl 0197.38601 [15] Sjögren, P, Harmonic spaces associated with adjoints of linear elliptic operators, Ann. inst. Fourier (Grenoble), 25, 509-518, (1975) · Zbl 0303.35034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.