The Cauchy integral and analytic continuation

One of the most important concepts on the theory of approximation by analytic functions is that of analytic continuation. In a typical problem, for example, there is generally a region Q, a Banach space B of functions analytic on Ω and a subfamily ⊂ B, each member of which is analytic on some larger open set, and one might be asked to decide whether or not is dense on B. It often happens, however, that either is dense or the only functions which can be so approximated have a natural analytic continuation across ϑΩ. A similar phenomenon is also known to occur even for approximation on sets without interior. In this article we shall describe a method for proving such theorems which can be applied on a variety of settings and, in particular, to: (1) the Bernštein problem for weighted polynomial approximation on the real line; (2) the completeness problem for weighted polynomial approximation on bounded simply connected regions; (3) the Shapiro overconvergence problem for sequences of rational functions with sparse poles; (4) the Akutowicz-Carleson minimum problem for interpolating functions. Although we shall present no new results, the method of proof, which is based on an argument of the author [6], seems sufficiently versatile to warrant exposition.