Testing analyticity on rotation invariant families of curves

Abstract:

Let $\Gamma \subset C$ be a piecewise smooth Jordan curve, symmetric with respect to the real axis, which contains the origin in its interior and which is not a circle centered at the origin. Let $\Omega$ be the annulus obtained by rotating $\Gamma$ around the origin. We characterize the curves $\Gamma$ with the property that if $f \in C(\Omega )$ is analytic on $s\Gamma$ for every $s$, $|s| = 1$, then $f$ is analytic in Int $\Omega$.