A function algebra providing new Mergelyan type theorems in several complex variables

Abstract The celebrated theorem of Mergelyan states that, if K is a compact subset of the complex plane with connected complement, then every continuous function on K which is holomorphic on its interior can be uniformly approximated on K by polynomials. This paper is concerned with polynomial and rational approximation in several complex variables, where the situation is much more complicated and far from being understood. In particular, we introduce a natural function algebra which allows us to obtain new Mergelyan type theorems for certain graphs as well as for Cartesian products of an arbitrary (possibly infinite) indexed family of planar compact sets. Finally, we identify a mistake in a classical result from 1969 and correct it within the framework of our new algebra.