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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.
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
Full Text: DOI Link
[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
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