Approximation by generalized Faber series in Bergman spaces on finite regions with a quasiconformal boundary (Q2564364)

Let \(G\) be a simple connected bounded domain with quasiconformal boundary. Let \(F_n\) be the \(n-th\) Faber polynomial of \(G\). Given a function \(f\) in the Banach space \(A^2(G)\), the author defines a series \(\sum_{n=1}^{\infty}a_n(f)F'_n(z)\) and shows that it converges uniformly on compact subsets of \(G\). Moreover, if \(S_n (f,z) := \sum_{k=1}^{k=n} a_k(f) F'_k (z)\) then \(|f-S_n(f,\cdot)|_{A^2(G)} \leq \sqrt {6n/(1-k^2)} E_n(f,G)\), where \(E_n(f,G):=\inf\{|f-p|_{A^2(G)}\);\(p\) is a polynomial of degree \(\leq n \}\). As a corollary one concludes that the series \(\sum_{n=1}^{\infty} a_n(f)F'_n(z)\) converges to \(f\) in the norm \(|{\cdot } |_{A^2(G)}\).