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UNIFORM APPROXIMATION BY bHARMONIC FUNCTIONS JOHN T. ANDERSON
 

Summary: UNIFORM APPROXIMATION BY b­HARMONIC FUNCTIONS
JOHN T. ANDERSON
Abstract. The Mergelyan and Ahlfors-Beurling estimates for the Cauchy
transform give quantitative information on uniform approximation by rational
functions with poles off K. We will present an analogous result for an integral
transform on the unit sphere in C2 introduced by Henkin, and show how it can
be used to study approximation by functions that are locally harmonic with
respect to the Kohn Laplacian b.
1. Introduction
The primary tool in the study of rational approximation on compact subsets of the
plane is the Cauchy transform ^µ of a compactly supported, complex Borel measure
µ, defined by
^µ(z) =

dµ()
- z
.
The following facts about ^µ can be found in many sources (see, for example [3], [7],
[8], [17]): ^µ is finite a.e. with respect to Lebesgue measure m on the plane, vanishes
at , and satisfies

  

Source: Anderson, John T. - Department of Mathematics and Computer Science, College of the Holy Cross

 

Collections: Mathematics