Some remarks on approximation in several complex variables

Abstract

In [2], in order to correct a false Mergelyan-type statement given in [4] on uniform approximation on compact sets 𝐾 in C 𝑑 , the authors introduced a natural function algebra 𝐴 𝐷 (𝐾 ) which is smaller than the classical one 𝐴(𝐾 ). In the present paper we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set 𝐾. Finally, we introduce a notion of 𝑂-hull of 𝐾 and strengthen known results.