UNIFORM APPROXIMATION BY POLYNOMIAL, RATIONAL AND ANALYTIC FUNCTIONS

Let K and X be compact plane sets such that $K\subseteq X$. Let P(K) be the uniform closure of polynomials on K, let R(K) be the uniform closure of rational functions on K with no poles in K and let A(K) be the space of continuous functions on K which are analytic on int(K). Define P(X,K),R(X,K) and A(X,K) to be the set of functions in C(X) whose restriction to K belongs to P(K),R(K) and A(K), respectively. Let S0(A) denote the set of peak points for the Banach function algebra A on X. Let S and T be compact subsets of X. We extend the Hartogs–Rosenthal theorem by showing that if the symmetric difference SΔT has planar measure zero, then R(X,S)=R(X,T) . Then we show that the following properties are equivalent: (i)R(X,S)=R(X,T) ;(ii)$S\setminus T\subseteq S_0(R(X,S))$ and $T\setminus S\subseteq S_0(R(X,T))$;(iii)R(K)=C(K) for every compact set $K \subseteq S\Delta T$;(iv)$R(X,S \cap \overline {U})=R(X,T \cap \overline {U})$ for every open set U in ℂ ;(v)for every p∈X there exists an open disk Dp with centre p such that We prove an extension of Vitushkin’s theorem by showing that the following properties are equivalent: (i)A(X,S)=R(X,T) ;(ii)$A(X,S \cap \overline {D})=R(X,T \cap \overline {D})$ for every closed disk $\overline {D}$ in ℂ ;(iii)for every p∈X there exists an open disk Dp with centre p such that