Uniform Approximation and Maximal Ideal Spaces

Let X be a compact set in the z-plane. We are interested in two function spaces associated with X: C(X) — space of all continuous complex-valued functions on X. P(X) =space of all uniform limits of polynomials on X. Thus a function ƒ on X lies in P{X) if there exists a sequence {Pn} of polynomials converging to ƒ uniformly on X. Clearly P(X) is part of C(X). QUESTION I. When is P(X) = C(X)t i.e. every continuous function approximable by polynomials? QUESTION I I . If P ( X ) ^ C ( X ) , how can we characterize those functions on X which lie in P(X)? The first man to pay any attention to these problems I believe was Weierstrass who in 1885 showed P(X) = C(X) when X is the unit interval on the real axis. If instead of the unit interval we consider an arbitrary Jordan arc (homeomorph of the unit interval), the problem is much harder. Walsh [ l ] proved