Finely holomorphic functions and quasi-analytic classes
Let \(C\{M_n\}\) be the class of all complex functions \(f: \mathbb{R}\to \mathbb{C}\), \(f\in C^\infty\), which satisfy the inequalities \(|f^{(n)} (x)|< \beta_f B_f M_n\) \((n=0,1, 2,\dots)\), where \(f^{(0)} =f\) and \(\beta_f\), \(B_f\) are constants depending on \(f\), while the constants \(M_n\) satisfy the following inequalities: \(M_0 =1\), \(M^2_n\leq M_{n-1} M_{n+1}\) \((n=1,2, \dots)\). A class \(C\{ M_n\}\) is said to be quasi-analytic if \(f^{(n)} (0)=0\) \((n=0, 1,2, \dots)\) imply \(f(x)=0\) \(\forall x\in \mathbb{R}\). Let \({\mathcal F}\) denote the class of complex functions \(f: \mathbb{R}\to \mathbb{C}\), for which there exists a fine domain \(U\) containing the real axis and a function \(\widetilde {f}\) finely holomorphic on \(V\) [cf. \textit{B. Fuglede}, Proc. 18th Scand. Congr. Math., Aarhus 1980, 22-38 (1981; Zbl 0462.30004)] that satisfies \(f(x)= \widetilde {f} (x)\) \(\forall x\in \mathbb{R}\). Let \(Q\) denote the class of complex functions \(f: \mathbb{R}\to \mathbb{C}\) for which there is a quasi-analytic class \(C\{ M_n\}\) containing \(f\). The author establishes that \(Q-{\mathcal F}\) as well as \({\mathcal F}-Q\) are non empty.
scientific article; zbMATH DE number 650842