Abstract

Let F be a relatively closed subset of an open set G of the plane. Alice Roth in [10] proved that if a function f is a uniform limit on F of holomorphic or meromorphic functions on G, it is possible to select the approximating functions m in such a way that the difference function f m can be extended continuously into the boundary of F. In this paper we consider the more general situation arising when one replaces ∂[bbar] by an elliptic operator p(D) with constant coefficients. We prove that the natural analog of Roth's theorem for the operator P(D) holds, at least for bounded open sets G. In those cases where one can find an inversion of the domain, preserving the uniform approximation by solutions of the operator, unbounded open sets are allowed too. This hannens when P(D)=δ and n=2. In this case we improve the main result of Goldstein and O w in [7], removing most oi the unnecessary conditions assumed in their work.

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