Inequalities for a perturbation theorem of Paley and Wiener

Abstract:

A classical theorem of Paley and Wiener states that the set of functions $\{ {e^{i{\lambda _n}t}}\} _{n = - \infty }^\infty$ forms a basis for ${L^2}( - \pi ,\pi )$ whenever the following condition is satisfied: \[ ( \ast )\quad ||\sum {{c_n}({e^{i{\lambda _n}t}} - {e^{int}})|{|^2} \leqq {\theta ^2} \sum {|{c_n}{|^2}} } \quad (0 \leqq \theta < 1).\] It is known that ($\ast$) holds whenever ${\lambda _n}$ is real and $|{\lambda _n} - n| \leqq L < \frac {1}{4}( - \infty < n < \infty )$, and may fail to hold if $|{\lambda _n} - n| = \frac {1}{4}$. In this note we show, more generally, that the condition $|{\lambda _n} - n| < \frac {1}{4}$ is also insufficient to ensure ($\ast$).