A Mergelyan-Vitushkin approximation theorem for rational modules

Abstract

Irob) + r1(z) g(z)), where each ri is a rational function with poles off X. In the cast in which g(z) = Z, the closures of 22(X, 2) in various norms were first considered by O’Farrell [4]. Later, several authors (e.g., Carmona, Trent, Verdera, and Wang) explored the subject. A question which arose from these investigations concerned the characterization of R(X, g), the uniform closure of 92(X, g) in C(X). When X has empty interior $ this was settled in [6] (also see [l]) by showing that R(X, g) = C(X) if and only if R(Z) = C(Z), where g is a smooth function and Z is the subset of X on which 8g vanishes. Here 8 = $(8/8x + id/L+) is the usual Cauchy-Riemann operator in the complex plane. The existence of interior points, however, makes the problem more -difficult. Note that a functionfsatisties a(Q,Y/ag) = 0 in an open set U if and only if f = h + gk with h and k holomorphic. Therefore it is natural to ask the following question: For an arbitrary compact set X, is