Approximation by polynomials in the mean with respect to area

This paper extends the concept of polynomial hull to arbitrary bounded sets of a complex plane, when an integral metric (relative to area) is considered. It is proved that for the completeness of the set of polynomials in a given closed or open set according to an integral metric, it is a necessary condition that the corresponding “polynomial hull” coincide with the interior of the set under consideration. The sufficiency of this condition is proved for various classes of sets. Using the notion of analytic $p$-capacity of sets, we obtain a full description of the compacta for which the polynomials are complete.
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