55.1 Runge’s Theorem
Consider the function,
defined on Ω ≡ B
is analytic on Ω.
Suppose you could approximate f
uniformly by polynomials on
a compact subset of Ω.
Then, there would exist a suitable polynomial
is a circle of radius
However, this is impossible because
= 1 while
shows you can’t expect to be able to uniformly approximate analytic functions on
compact sets using polynomials. This is just horrible! In real variables, you can
approximate any continuous function
on a compact set with a polynomial.
However, that is just the way it is. It turns out that the ability to approximate
an analytic function on Ω with polynomials is dependent on Ω being simply
All these theorems work for f having values in a complex Banach space. However, I
will present them in the context of functions which have values in ℂ. The changes
necessary to obtain the extra generality are very minor.
Definition 55.1.1 Approximation will be taken with respect to the following norm.