Density of the polynomials in Hardy and Bergman spaces of slit domains

Abstract

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that \(\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\}\) and with the property that the analytic polynomials are dense in the Bergman space \(\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma)\) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in \(H^{t}(\mathbb{D}\setminus\Gamma)\) ; improving upon a result in an earlier paper.