Strong Forelli property of Fréchet spaces and Alexander's theorem for Fréchet-valued formal power series
Abstract
We give sufficient conditions to ensure the convergence on some zero-neighbourhood in a Fréchet space E (resp. E = C^N) of a formal power series (resp. a sequence of formal power series) of Fréchet-valued continuous homogeneous polynomials provided that the convergence holds at a zero-neighbourhood of each complex line l_a := Ca for every a in A, a non-projectively-pluripolar set in E. The result in the case E = C^N is a Fréchet-valued analog of classical Alexander's theorem but under weaker assumptions. It is also shown that every Fréchet space has the strong Forelli property, i.e, for a non-projectively-pluripolar set A \subset \C^N, every Fréchet-valued function f on the open unit ball \Delta_N \subset \C^N, f \in C^\infty(0), such that its restriction on each complex line l_a, a in A, is holomorphic admits an extension to an entire function.