Uniqueness for non-harmonic trigonometric series

Abstract:

When $\lambda _{n} > 0$, $\lambda _{n} \uparrow \infty$ and \begin{equation*} \frac {1}{2}\left |a_{0}\right | +\sum _{n=1}^{\infty }\frac {\left |a_{n}\right |+\left |b _{n}\right |}{\lambda _{n}^{2}} < \infty , \end{equation*} if \begin{equation*} \frac {1}{2}a_{0}+\sum _{n=1}^{\infty }(a_{n}\cos \lambda _{n}x+b_{n}\sin \lambda _{n}x) = 0 \text {\quad everywhere $(-\infty , \infty )$}, \end{equation*} then \begin{equation*} a_{0}=a_{1}=b_{1}=\dots =a_{n}=b_{n}=\dots =0. \end{equation*} More generalized results are given.