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Eigenvalues of a specific Hankel matrix

I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by \begin{equation} G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2}, \end{...
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