Uniform rational approximation

Abstract:

Let K be a compact subset of the complex plane $\mathbb {C}$, and let $P(K)$ and $R(K)$ be the closures in $C(K)$ of polynomials and rational functions with poles off K, respectively. Suppose that $R(K) \ne C(K),\lambda$ is a nonpeak point for $R(K)$, and g is continuous on $\mathbb {C}$ and ${C^1}$ in a neighborhood of $\lambda$. Then $P(K)g + R(K)$ is not dense in $C(K)$. In fact, our proof shows that there are a lot of smooth functions which are not in the closure of $P(K)g + R(K)$.