Tính chất Forelli mạnh của các không gian Fréchet và định lý Alexander đối với các chuỗi luỹ thừa hình thức giá trị Fréchet

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.