Completeness of analytic functions and extremality of the coefficients of a Laurent series

We generalize Vitushkin's theorem on the fact that the completeness of the set of functions analytic on a compactum in the complex plane depends upon the extremality of the first coefficient of the Laurent series of the classes of functions connected with this compactum. We show that completeness is characterized by the extremality of the Laurent series coefficient with any fixed number $n$, $n\ge1$. The $n$-th analytic capacity considered, generalizing the concept of analytic capacity ($n=1$), also flexibly measures the set.