Complex Analysis (Princeton Lectures in Analysis, Volume II)

We know by Weierstrass’s theorem that any continuous function on a compact interval can be approximated uniformly by polynomials. With this result in mind, one may inquire about similar approximations in complex analysis. More precisely, we ask the following question: what conditions on a compact set K ⊂ C guarantee that any function holomorphic in a neighborhood of this set can be approximated uniformly by polynomials on K? An example of this is provided by power series expansions. We recall that if f is a holomorphic function in a disc D, then it has a power series expansion f(z) = ∑∞ n=0 anz n that converges uniformly on every compact set K ⊂ D. By taking partial sums of this series, we conclude that f can be approximated uniformly by polynomials on any compact subset of D. In general, however, some condition on K must be imposed, as we see by considering the function f(z) = 1/z on the unit circle K = C. Indeed, recall that ∫