Algebras of continuous functions on disks. (English) Zbl 0879.46025
The author continuous his research on the following approximation problem [see e.g., Proc. Am. Math. Soc. 97, 299-302 (1986; Zbl 0593.46045) and Indag. Math., New Ser. 6, No. 4, 477-479 (1995; Zbl 0840.30022)]:
For which functions \(g\) continuous in a small disk \(D\) centered at the origin in the complex plane the uniform closure of the polynomials in \(z^m\) and \(g\) coincides with \(C(D)\)?
As a typical result, we mention that this is true for \(m=2\) and \(g(z)= \overline z^2+z^3\). The proofs are based on a careful study of polynomial convex hulls in \(\mathbb{C}^n\).
For which functions \(g\) continuous in a small disk \(D\) centered at the origin in the complex plane the uniform closure of the polynomials in \(z^m\) and \(g\) coincides with \(C(D)\)?
As a typical result, we mention that this is true for \(m=2\) and \(g(z)= \overline z^2+z^3\). The proofs are based on a careful study of polynomial convex hulls in \(\mathbb{C}^n\).
Reviewer: R.Mortini (Metz)
MSC:
| 46J10 | Banach algebras of continuous functions, function algebras |
| 30E10 | Approximation in the complex plane |
| 32E20 | Polynomial convexity, rational convexity, meromorphic convexity in several complex variables |
| 41A10 | Approximation by polynomials |
