The Faber transform and analytic continuation

Abstract:

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