Density of the polynomials in Bergman spaces (Q1081729)

Let G be a bounded simply connected domain in the complex plane. Using a result of Hedberg, we show that the polynomials are dense in the Bergman space \(L^ 2_ a(G)\) if G is the image of the unit disc under a weak- star generator of \(H^{\infty}\). This result generalizes an old theorem (1934) of Farrell and Markusevic: the polynomials are dense in \(L^ 2_ a(G)\) if G is a Carathéodory domain. We also show that density of the polynomials in \(L^ 2_ a(G)\) implies density of the polynomials in the Hardy space \(H^ 2(G)\).