On zero sets of harmonic and real analytic functions

Abstract

In this paper we study some questions related to the zero sets of harmonic and real analytic functions in $${\mathbb {R}}^N$$ . We introduce the notion of analytic uniqueness sequences and, as an application, we show that the zero set of a non-constant real analytic function on a domain always has empty fine interior. We also prove that, for a certain category of sets $$E\subset {\mathbb {R}}^N$$ (containing the finely open sets), each function f defined on E is the restriction of a real analytic (respectively harmonic) function on an open neighbourhood of E if and only if f is “analytic (respectively harmonic) at each point” of E.