\\ \ell \, \geq \, 0}} 
\sum_{\substack{0 \leq m \leq N_j \\ 0 \leq q \leq N_\ell}} 
\overline{ \Big( B E_m^{n,n+j} \Big) (v')} 
\Big( A E_q^{n,n+\ell} \Big) (v) 
\langle \chi_m^{n,n+j},\chi_q^{n,n+\ell} \rangle \\
& \times \int_{\mathbb{T}} \big( -2\sqrt{k} \cos (\theta) + k + 1 - z \big)^{-1} 
\sin \big( (j + 1) \theta \big) \sin \big( (\ell + 1) \theta \big) \, d\theta.
\end{split}$$ Then, to complete the proof of the theorem, it remains only to show that $$\label{eq4,15}
\begin{split}
\frac{1}{\pi} \int_{\mathbb{T}} & \big( -2\sqrt{k} \cos (\theta) + k + 1 - z \big)^{-1} 
\sin \big( (j + 1) \theta \big) \sin \big( (\ell + 1) \theta \big) \, d\theta \\
& = \frac{1}{2\sqrt{k}} \sqrt{\frac{2}{\pi}} \left( - \frac{i e^{i(j + \ell + 2)\Phi}}{\sqrt{u}
\sqrt{4 - u}} + \frac{i e^{i\vert j - \ell \vert\Phi}}{\sqrt{u} \sqrt{4 - u}} \right),
\end{split}$$ where the relation between $z$, $u$ and $\Phi$ is given by . To do this, we have to deal with the discrete Fourier transform $\mathcal{F} : \ell^2({{\mathbb{Z}}},{{\mathbb{C}}}) 
\rightarrow L^2({{\mathbb{T}}})$, defined for any $x \in \ell^2({{\mathbb{Z}}},{{\mathbb{C}}})$ and $f \in L^2({{\mathbb{T}}})$ by $$(\mathcal{F}x)(\theta) := (2\pi)^{-\frac{1}{2}} \sum_{n \, \in \, {{\mathbb{Z}}}}
e^{-in\theta} x(n), \quad
\big( \mathcal{F}^{-1} f \big)(n) := (2\pi)^{-\frac{1}{2}} 
\int_{{{\mathbb{T}}}} e^{in\theta} f(\theta) \, d\theta.
$$ 