On a theorem of Ingham on nonharmonic Fourier series

Abstract

A well-known result due to Ingham [3] shows that the system of complex exponentials { e i λ n t } \{ {e^{i\lambda _{n}t}}\} is a basic sequence in L 2 ( − π , π ) {L^2}( - \pi ,\pi ) whenever λ n + 1 − λ n ⩾ γ > 1 {\lambda _{n + 1}} - {\lambda _n} \geqslant \gamma > 1 . In this note, we show that the system need not be basic if λ n + 1 − λ n > 1 {\lambda _{n + 1}} - {\lambda _n} > 1 .