POLYNOMIAL AND RATIONAL APPROXIMATION IN BANACH FUNCTION ALGEBRAS

Let K and X be compact plane sets such that K Í X. We define P(X,K) = {f ϵ C(X) : f|K ϵ P(K)}, R(X,K) = {f ϵ C(X) : f|K ϵ R(K)} where P(K) and R(K) are uniform closures of polynomials and rational functions with poles off K, respectively. Let S0(A) denote the set of peak points of the Banach function algebra A on X. Let S and T be compact subsets of the compact plane set X. We first show that if the symmetric difference S + T has planar measure zero then R(X, S) = R(X, T), which implies the Hartogs - Rosenthal theorem. Then we show that the following properties are equivalent:(i) R(X, S) = R(X, T). (ii) S T Í S0(R(X, S)) and T S Í S0(R(X, T)). (iii) R(K) = C(K) for every compact subset K Í X (SÇ T). Moreover, P(X, S) = P(X, T) if and only if S T Í S0(P(X, S)) and T S Í S0(P(X, T)). Finally, we show that some of the above properties are satisfied for the extended Lipschitz algebras.

کلیدواژه ها:

Uniform algebras ، polynomial and rational approximation ، peak points ، extended Lipschitz algebras