No abstract

No abstract

ABSTRACT.Let X be a compact subset of C with empty interior and let g be a complex function of class C2 in a neighborhood of X. For Z = {z £ X\dg(z)/dz = 0}, we prove that R(X) + gR(X) is uniformly dense in C(X) if and only if R(Z) = C(Z).If X is a compact subset of C, we denote by Ro(X) the algebra of rational functions with poles off X, and by R(X) the uniform closure of Ro(X) in C(X). We use the symbol d for the differential operator d/dz = ^(d/dx + id/dy).The purpose of this paper is to present a proof of the following result:THEOREM. Suppose thatX is a compact subset ofC with empty interior, and that g is a complex function twice continuously differentiable on a neighborhood of X. Write Z = {x G X | dg(z) = 0}. Then, Ro(X) + gRo(X) is uniformly dense in C(X) if and only if R(Z) = C(Z).COROLLARY. If X is a compact subset of C with no interior, and if h is an entire function, then Ro(X) + hRo{X) is uniformly dense in C(X).The particular case of the above corollary in which h(z) = z, z E C, was conjectured in O'Farrell [3] and Wang [6], and it has been recently proved by Trent and Wang [5]. Rational modules of type Ro(X) + Ro(X)z had been introduced by O'Farrell [3] in connection with problems of rational approximation in Lipschitz norms.The basic tool used in the proof of the theorem is an integral formula for infinitely differentiable functions with compact support disjoint from Z (1). In a second step we introduce an integral kernel, associated to g, and the corresponding transforms of compactly supported measures. Using this transform, we are able to prove that a measure on X, orthogonal to Ro(X)+gRo(X) must be supported on Z, and then, the desired conclusion follows via standard arguments.We proceed now to state and prove several results which will be needed in the proof of the theorem.We denote by m the Lebesgue measure on C, and by C2(U) (resp. D2(U)) the set of complex functions (resp. compactly supported functions) twice continuously differentiable on the open subset U of C.

No abstract

No abstract