Some results on spherical approximation (Q2865912)

Let \(D=\{z\in\mathbb{C};\;| z| <1\}\) be the unit disc. Recently the second author showed that the set of uniform limits with respect to the chordal metric \(\chi\) of polynomials on \(\overline{D}\) consists of all the continuous functions \(f:\overline{D}\rightarrow \mathbb{C}\cup\{\infty\}\) such that either \(f|_D\equiv \infty\) or \(f(D)\subset\mathbb{C}\) with \(f|_D\) holomorphic [Bull. Lond. Math. Soc. 44, No. 4, 775--788 (2012; Zbl 1261.30007)]. This statement can be viewed as a version of Mergelyan's theorem in case of the chordal metric. Recall that the classical Mergelyan theorem extends to compact sets with finite connectedness; then polynomials are replaced by rational functions with prescribed poles. So the authors obtain the following extension of the above result: let \(\Omega\subset \mathbb{C}\) be a bounded open set whose boundary consists of a finite set of pairwise disjoint Jordan curves, \(A\) be a set containing one point from each component of \((\mathbb{C}\cup\{\infty\})\setminus\overline{\Omega}\) and \(f:\overline{\Omega}\rightarrow \mathbb{C}\cup\{\infty\}\) be a continuous function. Then, the function \(f\) is the \(\chi\)-uniform limit on \(\overline{\Omega}\) of a sequence of rational functions with poles included in \(A\) if and only if for each component \(V\) of \(\Omega\) either \(f|_V\equiv\infty\) or \(f(V)\subset\mathbb{C}\) with \(f|_V\) holomorphic.\vskip1mm Finally passing from compact sets to unbounded closed sets the authors introduce the notion of \(\chi\)-Arakelian sets in \(\mathbb{C}\). Several questions remain open.