Uniform rational approximation

Abstract

Let K be a compact subset of the complex plane C \mathbb {C} , and let P ( K ) P(K) and R ( K ) R(K) be the closures in C ( K ) C(K) of polynomials and rational functions with poles off K, respectively. Suppose that R ( K ) ≠ C ( K ) , λ R(K) \ne C(K),\lambda is a nonpeak point for R ( K ) R(K) , and g is continuous on C \mathbb {C} and C 1 {C^1} in a neighborhood of λ \lambda . Then P ( K ) g + R ( K ) P(K)g + R(K) is not dense in C ( K ) 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 ) P(K)g + R(K) .