Ingham Inequality

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

t_{j+1}-t_{j}\geq\gamma>1,\quad 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}\leq\int_{-\pi}^{\pi}\left|\sum_{j=-n}^{n}\sqrt{% \frac{1}{2\pi}}c_{j}e^{it_{j}x}\right|^{2}dx,

where

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