×

zbMATH — the first resource for mathematics

Pervasive algebras of analytic functions. (English) Zbl 0981.46044
Let \(X\) be a compact Hausdorff space and \(S\) a complex or real closed subspace of \(C(X,\mathbb{C})\) or \(C(X,\mathbb{R})\) respectively, and let \(Y\) be a closed subset of \(X\). \(S\) is said to be complex or real pervasive on \(Y\) if the functions of \(S\) restricted to \(E\) are dense in \(C(E,\mathbb{C})\) or \(C(E,\mathbb{R})\), respectively for each proper closed subset \(E\) of \(Y\). These properties are investigated for the case where \(X\) is an open proper subset \(U\) of the Riemann sphere \(\widehat{\mathbb{C}}\) and \(S\) is the algebra \(A(U)\) of all complex valued functions continuous on \(\widehat{\mathbb{C}}\) and analytic on \(U\), or \(S= \text{Re }A(U)\), respectively, and it is supposed that \(U\) has no inessential boundary points – i.e. points where all functions of \(A(U)\) can be extended analytically. Then \(A(U)\) is complex pervasive on \(\partial U\) if and only if \(\partial U_i=\partial U\) for each component \(U_i\) of \(U\). If \(\text{Re }A(U)\) is real pervasive on \(\partial U\) then \(U\) has at most one component \(U_k\) that is not simply connected, and in this case \(\partial U_k=\partial U\). If on the other hand \(U\) has at least one component \(U_k\) such that \(\partial U_k=\partial U\) then \(\text{Re }A(U)\) is real pervasive on \(\partial U\). In case that all components \(U_i\) of \(U\) are simply connected and \(\partial U_i\neq\partial U\), then the real pervasiveness is characterized in terms of properties of the boundary points.
MSC:
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Browder, A., Introduction to function algebras, (1969), W. A. Benjamin New York · Zbl 0199.46103
[2] Carleson, L., Selected problems on exceptional sets, (1967), Van Nostrand Princeton
[3] Cerych, J., A word on pervasive function spaces, Complex analysis and applications, Varna, 1981, (1984), p. 107-109 · Zbl 0594.46050
[4] Davie, A., Analytic capacity and approximation problems, Trans. amer. math. soc., 171, 409-444, (1971) · Zbl 0263.30032
[5] Gamelin, T., Uniform algebras, (1969), Prentice-Hall Englewood Cliffs · Zbl 0213.40401
[6] Gamelin, T.; Garnett, J., Pointwise bounded approximation and Dirichlet algebras, Functional anal., 8, 360-404, (1971) · Zbl 0223.30056
[7] Hoffman, K.; Singer, I.M., Maximal algebras of continuous functions, Acta math., 103, 217-241, (1960) · Zbl 0195.13903
[8] Netuka, I., Pervasive function spaces and the best harmonic approximation, J. approx. theory, 51, 175-181, (1987) · Zbl 0641.41025
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.