Approximation in Lipschitz algebras

Abstract

When X is a compact subset of C n and 0 < α ≤ 1, we define subalgebras of the Lipschitz algebras Lip(X, α) and lip(X, α), which are generated by polynomials, rational functions with poles off X, functions which are analytic in some neighbourhood of X, or functions which are analytic in the interior of X. We investigate their maximal ideal spaces and the equality among them. For certain compact plane sets X, we extend a result due to Dales and Davie, by showing that every continuously differentiable function on X can be approximated by rational functions in lip(X, α) norm. Finally we show that the algebra of rational functions with poles off X is dense in lip(X, α) on any compact plane set X, with planar measure zero, and then extend this result for particular compact subsets of C n , n > 1.