Polynomial Approximation in Bers Spaces of Carathéodory Domains

Let D be a Carathéodory domain and let t D = Sup { q ∈ R : μ q ( D ) = ∞}, where μ q ( D ) =∬ Dλ D 2 − q d x d y .Here λ D ( Z ) is the Poincaré metric for D . Then 1 ⩽ t D ⩽ 2. Let I ( t D = { q ∈ R : μ q ( D ) < ∞}. Then I ( t D ) = [ t D , ∞ ) if μ t D( D ) < ∞ , and I ( t D ) = ( t D , ∞) if μ t D( D ) = ∞ . Define B q p ( D ) , the Bers space, to be the class of analytic functions f in D , such that‖ f ‖ pq , p = ∬ D| f | p λ D2 − q d x d yis finite, 0 < p < ∞, q ∈ I ( t D ). It is well known that the polynomials are dense in B q p ( D ) for q ⩾ 2. We show that they are dense in B q p ( D ) for any q ∈ I ( t D ) and 0 < p < ∞. Other related results are discussed. This extends our previous work for the case t D = 1.