A generalization of Szegö's theorem (Q1362397)

The paper deals with a generalization of Szegö's theorem on the completeness of the family \(\{e^{inx}\}^\infty_{k=0}\) in the weighted Gevrey space \(I_\alpha(d\mu)\) \((\alpha>1)\), which consists of infinitely differentiable \(2\pi\)-periodic functions satisfying the condition \[ |f|_{I_\alpha(d\mu)}= \Biggl(\sum^\infty_{n=0} {1\over 2\pi} \int^\pi_{-\pi} \Biggl|{f^{(n)}(x)\over (n!)^\alpha}\Biggr|^2 d\mu(x)\Biggr)^{{1\over 2}}<\infty. \]