The stability of the completeness and minimality in $L^2$ of a system of exponential functions

A. M. Sedletskii

Matematičeskie zametki, Tome 15 (1974) no. 2, pp. 213-219

Résumé

Let the sequences $\{\lambda_n\}$ and $\{\alpha_n\}$ of complex numbers satisfy the conditions: 1) $\sup|\operatorname{Im}\lambda_n|=h<\infty$; 2) the number of points $\lambda_n$ in the rectangle $|t-\operatorname{Re}z|\le1$, $|\operatorname{Im}z|\le h$ is uniformly bounded with respect to $t\in(-\infty,\infty)$; 3) $\{\alpha_n\}\in l^p$ for some $p<\infty$. Then the systems $\{\exp(i\lambda_nx)\}$ and $\{\exp(ix(\lambda_n+\alpha_n))\}$ are simultaneously complete or noncomplete (minimal or nonminimal) in $L^2(-a,a)$ ($a<\infty$).