Approximation by rational modules in Sobolev and Lipschitz norms

Abstract

Let X⊂ C be compact, 0> n∈ Z, and g a continuous function on X. Let R( n, g, X) be the rational module consisting of the functions on X of the type r 0 + r 1 g + ··· + r n g n , where r j is a rational function with poles off X, 0 ⩽ j ⩽ n. It is shown that if X is nowhere dense, g is sufficiently smooth, and \\ ̄ t6g(z) ≠ 0, z ∈ X , then the restriction to X of each function in C∈( C) is approximable in the Lip( n − 1, X)-norm, n ⩾ 2, by functions in R( n, g, X). Also dealt with are approximation problems in Sobolev norms by more general types of rational modules.