×

Function algebras on disks. II. (English) Zbl 1134.46031

The authors give sufficient conditions on functions \(g\) being \(C^1\) on a small disk \(D\) around the origin in order that the algebra \([z^2,g^2,D]\) generated by \(z^2\) and \(g^2\) is dense in the space of all continuous, complex valued functions on \(D\). Previous results of this type were given mainly by P.de Paepe [Math.Z.212, No.2, 145–152 (1993; Zbl 0789.30027)] and by P.de Paepe and the first author of this paper in the first part of this series [Complex Variables, Theory Appl.47, No.5, 447–451 (2002; Zbl 1028.46081)]. A novelty here is that the lowest order terms of \(g(z)-\overline z\) are linear combinations of monomials \(z^m\overline z^n\), where \(m+n\geq 0\), \(m,n\in\mathbb Z\).

MSC:

46J10 Banach algebras of continuous functions, function algebras
30H05 Spaces of bounded analytic functions of one complex variable
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexander, H.; Wermer, J., Several Complex Variables and Banach Algebras, (Graduate Texts in Math., vol. 35 (1998), Springer-Verlag: Springer-Verlag New York) · Zbl 0894.46037
[2] Henkin, G.; Leiterer, I., Theory of Functions on Complex Manifolds, (Monographs Math., vol. 79 (1984), Birkhäuser: Birkhäuser Boston)
[3] Hörmander, L., An Introduction to Complex Analysis in Several Variables, (North-Holland Math. Library, vol. 7 (1990), North-Holland: North-Holland Amsterdam) · Zbl 0138.06203
[4] Kallin, E., Fat polynomially convex sets, (Function Algebras, Proceedings of International Symposium on Function Algebras, Tulane Univ, (1965) (1966), Scott Foresman: Scott Foresman Chicago), 149-152 · Zbl 0142.10104
[5] Dieu Nguyen, Quang, Local polynomial convexity of tangential unions of totally real graphs in \(C^2\), Indag. Math., 10, 349-355 (1999) · Zbl 1027.32015
[6] Dieu Nguyen, Quang, Local hulls of unions of totally real graphs lying in real hypersurfaces, Michigan Math. Journal, 47, 2, 335-352 (2000) · Zbl 0994.32011
[7] Dieu Nguyen, Quang; de Paepe, R. J., Function algebras on disks, Complex Variables, 47, 447-451 (2002) · Zbl 1028.46081
[8] O’Farrell, A. G.; Preskenis, K. J.; Walsh, D., Holomorphic approximation in Lipschitz norms, Contemp. Math., 32, 187-194 (1984) · Zbl 0553.32015
[9] de Paepe, P. J., Algebras of continuous functions on disks, Proc. Roy. Acad. Irish Sect. A, 96, 85-90 (1996) · Zbl 0879.46025
[10] de Paepe, P. J., Approximation on a disk I, Math. Z., 212, 145-152 (1993) · Zbl 0789.30027
[11] de Paepe, P. J., Eva Kallin’s lemma, Bull. London Math. Soc., 33, 1-10 (2001) · Zbl 1041.32006
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.