Analytic continuation from a family of lines (Q948870)

Let \(\Lambda \subset {\mathbb R^2}\) be a \(C^2\)-smooth convex curve with strictly positive curvature, let \(l_{\lambda}\) be the tangent line to \(\Lambda\) at \(\lambda \in \Lambda.\) The main result of this paper is Theorem 1. Let \(f\) be a complex function in the exterior of \(\Lambda.\) Let there be for every \(\lambda \in \Lambda\), an entire function \(f_{\lambda}\) on \(\mathbb C = \mathbb R^2\) such that \(f| _{l_{\lambda}} = f_{\lambda}| _{l_{\lambda}}\). If the map \((z, \lambda) \to f_{\lambda} (z)\) is continuous, then \(f\) extends as an entire function on \(\mathbb C^2\). In Theorem 6 it is shown that this fact is true without the requirement of continuity of the map \((z, \lambda) \to f_{\lambda} (z)\) in the case when \(\Lambda \) is a real-analytic convex curve with strictly positive curvature.