FINELY HOLOMORPHIC-FUNCTIONS AND QUASI-ANALYTIC CLASSES

We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other. Let 2 denote the set of complex functions f: R --> C for which there exists a quasi-analytic class C{M(n)} containing f Let F denote the set of complex functions f: R --> C for which there exist a fine domain U containing the real line R and a function f finely holomorphic on U satisfying f(x) = f(x) for all x is-an-element-of R. The ''power'' of unique continuation is incomparable in these two cases (2\F is non-empty F\2 is non-empty).