Abstract

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 \textit{P. de Paepe} [Math. Z. 212, No. 2, 145–152 (1993; Zbl 0789.30027)] and by \textit{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 \(m+n\geq 0\), \(m,n\in\mathbb Z\).