Pointwise bounded approximation by polynomials. (English) Zbl 0768.30024
Let \(U\) be a bounded open set, and let \(F\) be a closed subset of \(\overline U\). If for each function \(f\), analytic and bounded in \(U\), and each \(\varepsilon>0\) there exists a sequence of polynomials \(p_ m\) such that \(p_ m(z)\to f(z)\) for \(z\in U\),
\[
\sup_{z\in U} | p_ m(z)| \leq \sup_{z\in U}| f(z)|, \qquad \sup_{z\in F} | p_ m(z)|\leq \sup_{z\in F\cap U} | f(z)|+ \varepsilon,
\]
then the authors call \((U,F)\) a Farrell pair. They characterize Farrell pairs and extend a result of A. Stray [Pac. J. Math. 51, 301-305 (1974; Zbl 0288.30035)].
Reviewer: K.Menke (Dortmund)
MSC:
30E10 | Approximation in the complex plane |
Keywords:
Farrell pairsCitations:
Zbl 0288.30035References:
[1] | Stray, Pacific J. Math. 51 pp 301– (1974) · Zbl 0288.30035 · doi:10.2140/pjm.1974.51.301 |
[2] | Fisher, Function Theory on Planar Domains (1983) · Zbl 0511.30022 |
[3] | Gamelin, Uniform Algebras (1969) |
[4] | DOI: 10.1112/plms/s3-51.2.369 · Zbl 0573.30029 · doi:10.1112/plms/s3-51.2.369 |
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.