Analyticity on rotation invariant families of curves

Abstract:

Let $\mathfrak {G}$ be a rotation invariant family of smooth Jordan curves contained in $\Delta$, the open unit disc in ${\mathbf {C}}$. For each $\Gamma \in \mathfrak {G}$ let ${D_\Gamma }$ be the simply connected domain bounded by $\Gamma$. We present various conditions which imply that if $f$ is a continuous function on $\Delta$ such that for every $\Gamma \in \mathfrak {G}$ the function $f|\Gamma$ has a continuous extension to $\overline {{D_\Gamma }}$ which is analytic in ${D_\Gamma }$, then $f$ is analytic in $\Delta$.