Given $\alpha \in \mathbb{R}$ and $N \geq 1$, I hope that the matrix $$A_N := (e^{i \alpha (k-l)})_{1 \leq k,l \leq N}$$ admits a simple reduction $A_N = U^* D U$ with $U^* U = I_N$ and $D$ a diagonal matrix (notation $U^* = \overline{U^{\intercal}}$).
Indeed, $A_N$ is hermitian, so such a reduction exists. My question is : is there a simple expression of $U$ (for all $\alpha \in \mathbb{R}$) ?
I tried to look at circular matrix, but $A_N$ isn't that type of matrix. I suspect this matrix to have a name, but I couldn't find it. I would be happy if you have any hint or reference.
For the context, I would like to diagonalize the matrix $\Big(\int_{\mathbb{R}} e^{i \alpha (k-l)} g(\alpha) d\alpha\Big)_{1 \leq k, l \leq N}$ with $g$ a nice and positive function.