Some results on spherical approximation

We extend Mergelyan's theorem to the case of compact sets K ⊂ℂ bounded by a finite number of disjoint Jordan curves, where the approximation is uniform with respect to the chordal metric χ and it is realized by rational functions with prescribed poles off K . Allowing poles in K °, as well, we obtain an analog result with the only assumption that K c has a finite number of components. We also obtain a Runge's‐type result and we introduce the notion of χ‐Arakelian sets in ℂ. In this respect, we prove that the real line is such a set.