Real and Complex Analysis (Rudin)

390 REAL AND COMPLEX ANALYSIS
Put F = f1 + f2 + f3 + .... By (4), the series converges to f on K, and it
converges uniformly on X. Hence F is continuous. Also, the support of Flies
~W W
Mergelyan's Theorem
20.5 Theorem If K is a compact set in the plane whose complement is connected,
iff is a continuous complex function on K which is holomorphic in the
interior of K, and if e > 0, then there exists a polynomial P such that
I f(z) - P(z) I < efor all z E K.
If the interior of K is empty, then part of the hypothesis is vacuously satisfied,
and the conclusion holds for every f E C(K). Note that K need not be connected.
PROOF By Tietze's theorem, f can be extended to a continuous function in
the plane, with compact support. We fix one such extension, and denote it
again byf.
For any ~ > 0, let ro(~) be the supremum of the numbers
where Z1 and Z2 are subject to the condition I Z2 - z11 S;~. Since f is uniformly
continuous, we have
lim ro(~) = 0. (1)
6 .... 0
From now on, ~ will be fixed. We shall prove that there is a polynomial
P such that
I f(z) - P(z) I < 1O,OOOro(~) (z E K). (2)
By (1), this proves the theorem.
Our first objective is the construction of a function E C~(R2), such that
for all z
and
I f(z) - (z) I S; ro(~), (3)
I (13Xz) I < 2i~) , (4)
(5)