Mergelyan's approximation theorem for rational modules
type: Item
instance of: scholarly article
title: Mergelyan's approximation theorem for rational modules
zbMATH Open document ID: 0574.30041
DOI: 10.1016/0021-9045(85)90074-7
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: 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 \({\mathbb{C}}\).
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 3916680
zbMATH Keywords: rational functions
zbMATH Keywords: uniform closure
zbMATH Keywords: rational approximation
zbMATH Keywords: Cauchy- Riemann operator
zbMATH Keywords: Mergelyan approximation theorem
author: Joan Josep Carmona
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90074-7
OpenAlex ID: W1999712246
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description: scientific article; zbMATH DE number 3916680