Excesses of systems of exponential functions

A. M. Sedletskii

Izvestiya. Mathematics , Tome 16 (1981) no. 1, pp. 191-205

Résumé

A nonnegative sequence $\{\alpha_n\}$ is called an admissible majorant if the condition $|\lambda_n-\mu_n|\leqslant\alpha_n$, where $\{\lambda_n\}$ and $\{\mu_n\}$ are real regular sequences, implies that the systems of functions $\{\exp(i\lambda_nx)\}$ and $\{\exp(i\mu_nx)\}$ have the same excess in the space $L^2(-a,a)$ ($a\infty$). A complete characterization of admissible majorants is given for the class of sequences $\alpha_n\downarrow0$ that have the smoothness property $\alpha_{n+1}\sim\alpha_n$. This is used to establish the definitiveness of the author's criterion for the stability of the excess of a system of exponentials in $L^2$. Bibliography: 10 titles.