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.

Mergelyan著名的定理指出，如果K是具有连接补码的复平面的紧凑子集，则K上内部是全纯的每个连续函数都可以在K上统一近似。通过多项式。本文涉及几个复杂变量的多项式和有理逼近，情况复杂得多，人们对此尚不了解。特别是，我们引入了一种自然函数代数，它使我们能够为某些图以及任意（可能是无限）索引平面紧凑集的笛卡尔乘积获得新的Mergelyan型定理。最后，我们从1969年的经典结果中识别出一个错误，并在新代数的框架内对其进行更正。