Ingham Inequality

Let ${(t_j)}_{j\in \mathbb{Z}}$ a increasing sequence of positive real numbers such that

$$t_{j+1}-t_j\ge \gamma >1,j\in \mathbb{Z}.$$

Then for all $n\in \mathbb{N}$ and for all complex sequences ${(c_j)}_{j=-n}^n$, we have

$$m\sum _{j=-n}^n{|c_j|}^2\le {\int }_{-\pi }^{\pi }{\left|\sum _{j=-n}^n\sqrt{\frac{1}{2\pi }}{c}_j{e}^{i{t}_{j}x}\right|}^2𝑑x,$$

where

$$m=\frac{2}{\pi }\left(1-\frac{1}{{\gamma }^2}\right).$$