 
Summary: Approximation Problems
on the Unit Sphere in C
2
John T. Anderson and John Wermer
Dedicated to the memory of S. Ya. Khavinson
Abstract. Let be the graph of a Holder continuous function over a Swiss
cheese E contained in the open unit disk and having the property that every
Jensen measure for R(E) is trivial. We show that if lies in the boundary of
the unit ball in C 2 , then R() = C(). In the appendix we give a geometric
interpretation of a class of sets X on the sphere introduced by R. Basener,
for which R(X) 6=
C(X).
1. Introduction
Let X be a compact subset of C n . We denote by R 0 (X) the algebra of all functions
P=Q where P and Q are polynomials on C
n and Q 6= 0 on X , and we denote by
R(X) the uniform closure of R 0 (X) in the space C(X) of continuous functions on
X . We are interested in nding conditions on X that imply that R(X) = C(X),
i.e., that each continuous function on X is the uniform limit of a sequence of
rational functions holomorphic in a neighborhood of X .
