The Faber transform and analytic continuation

Source abstract

Let Ω ⊆ C \Omega \subseteq C be a bounded, simply connected domain, and let { Φ n ( w ) } n = 0 ∞ \left \{ {{\Phi _n}\left ( w \right )} \right \}_{n = 0}^\infty be the Faber polynomials associated with Ω \Omega . Given f ( z ) = ∑ k = 0 ∞ c k z k f\left ( z \right ) = \sum \nolimits _{k = 0}^\infty {{c_k}{z^k}} analytic in Δ ( 0 , 1 ) \Delta \left ( {0,1} \right ) we consider the function $$F ( w ) = ∑ k = 0 ∞ c k Φ k ( w ) . F\left ( w \right ) = \sum \limits _{k = 0}^\infty {{c_k}{\Phi _k}\left ( w \right )} .$$ We show that with proper restrictions on ∂ Ω \partial \Omega , the existence of an analytic continuation of f f across a subarc of C ( 0 , 1 ) C\left ( {0,1} \right ) implies the existence of an analytic continuation of F F across a subarc of ∂ Ω \partial \Omega . Some converse results are also established.