APPROXIMATION BY HOLOMORPHIC-FUNCTIONS IN L(P) SPACE, LIP ALPHA SPACE AND BMO SPACE

Vitushkin-type theorems on the approximation by holomorphic functions in the complex plane are established. More precisely, let F be a closed (or measurable) subset of the complex plane and let B be any one of the following spaces of functions defined on F: L(p)(F), 1 < p < infinity, Lip-alpha(F), 0 < alpha < 1, BMO(F), or C(m)(F). Let A(B) be the set of those functions in B which are holomorphic on the interior of F. We characterize, in terms of appropriate capacities, those sets F for which every function in A(B) can be approximate, in the B-norm on F, by functions holomorphic in a neighbourhood of F. Our argument is along the lines of the original approximation scheme of A. G. Vitushkin and generalizes, to unbounded sets F, results of many authors, including T. Bagby, P. Lindberg, A. G. O'Farrell and J. Verdera. It should be noted that similar results for the uniform norm have been known for some time.