Approximate Carleman theorems and a Denjoy-Carleman maximum principle
Let \((M_n)_{n\geq 0}\) be a positive sequence satisfying \(M_0=1\), \(M_n^2\leq M_{n-1}M_{n+1}\) \((n\geq 1)\) and \(\sum_{n=1}^{\infty}M_n^{-1/n}=\infty.\) The authors show that if \(f\) is in the Carleman class \(C_{\mathbb R}(M_n):=\{f\in C^\infty(\mathbb R): |f^{(n)}(x)| \leq c_f\rho_f^nM_n\), \(x\in\mathbb R\), \(n\geq 0\}\) and \(\limsup| f^{(n)}(0)| ^{1/n}\leq C\), then for all integers \(m,n\geq 0\) \[ \sup_{x\in\mathbb R}| f(n+m)(x)| \leq C^n\sup_{x\in\mathbb R}| f^{(m)}(x)| . \] If, in particular, \(C=0\) then one obtains a generalization of the well-known Denjoy-Carleman theorem (\(f^{(n)}=0\) for \(n=0,1,\dots \implies f(x)=0,\;x\in\mathbb R\)). As an application of the above ideas, the authors obtain a generalization of Carleman's theorem on the unique determination of probability measures by their moments.
scientific article; zbMATH DE number 2103669