Approximation on a disk. I
In the functions \(F\) and \(G\) are defined in a neighborhood of the origin in the complex plane and behave in a certain sense as \(z^ m\) and \(\overline z^ n\) with \(\text{gcd} (m,n)=1\), then on sufficiently small closed disks \(D\) around 0 every continuous function on \(D\) can be approximated uniformly on \(D\) by polynomials in \(F\) and \(G\). A new proof of this theorem is given. Our main result is an example showing that this approximation result is in general not valid for \(m=n=2\). In the process of constructing such an example one also comes across an example of two polynomially convex closed disks \(X_ 1\) and \(X_ 2\) in \(\mathbb{C}^ 2\) which meet at only one point, which are tangent at this point and are such that the union of the two disks is not polynomially convex. [For part II, see the following review].
scientific article; zbMATH DE number 549722
scientific article