Completeness and basis properties of systems of exponentials in weighted spaces \(L^{p}(-\pi,\pi)\) (Q2473759)

The author studies the system of exponentials \(e(\Lambda)= \{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}\), where \(\lambda_n= n+({1+\alpha\over p}+ l(|n|))\text{sign\,}n\), \(l(t)\) is a slowly varying function, and \(l(t)\to 0\), \(t\to\infty\). The author proves the completeness criterion and a basis condition for the system \(e(\Lambda)\) in the weight spaces \(L^p(-\pi,\pi)\). This result is formulated in terms of generating function \(F(z)\) of the sequence \(\{\lambda_n\}\), where \[ F(z)= z\prod^\infty_{n=1} \Biggl(1-\Biggl({z\over \lambda_n}\Biggr)^2\Biggr). \] The proof is based on the estimate for the generating function \(F(z)\) of the sequence \(\{\lambda_n\}\) and some results of Sedletskii.