Mergelyan's approximation theorem for rational modules
The main results of this paper are the following: Theorem 1. Let X be a compact subset of \({\mathbb{C}}\) such that \({\mathbb{C}}-X\) has finitely many connected components. Suppose that g is a three-times continuously differentiable complex function in a neighbourhood of X, which satisfies \({\bar \partial}g(z)\neq 0\) for an \(x\in X\). Then each \(f\in C(X)\), satisfying \({\bar \partial}({\bar \partial}f/{\bar \partial}g)=0\) in \(\overset\circ X\) in the weak sense can be uniformly approximated on X by functions h satisfying \({\bar \partial}({\bar \partial}h/{\bar \partial}g)=0\) in a neighbourhood of X. Theorem 2. Let X be a compact subset of \({\mathbb{C}}\) with finitely many complementary components, and let n be a nonnegative integer. Then each \(f\in C(X)\) satisfying \({\bar \partial}^{n+1}f=0\) in \(\overset\circ X\) can be approximated by functions h satisfying \({\bar \partial}^{n+1}h=0\) in a neighbourhood of X. In the above results we have written \({\bar \partial}=(\partial /\partial x+i\partial /\partial y)\) for the usual Cauchy-Riemann operator in the complex plane \({\mathbb{C}}\). We can observe that the second theorem is the classical Mergelyan approximation theorem by polynomials when \(n=0\). Therefore the above results should be viewed as Mergelyan approximation theorems for the elliptic operators \({\bar \partial}({\bar \partial}/{\bar \partial}g)\) and \({\bar \partial}^{n+1}\). The proof of these theorems uses adequate reproducing formulae and a modification of the classical proof of Mergelyan's theorem. It is interesting to remark that it is not known if the above theorems remains true for every compact subset of \({\mathbb{C}}\).
scientific article; zbMATH DE number 3916680
scientific article