I'm in the context of discrete Fourier analysis and, in particular, trigonometric polynomial interpolation.
I cannot understand how to prove the following equality: $$ \frac{1}{2}\left(\hat f(-N/2+\ell N)e^{-iN/2x}+ \hat f(N/2+\ell N)e^{iN/2x}\right)+ \sum_{|k|<N/2}\sum_{\ell\neq 0} \hat f (k+\ell N)e^{ikx}\\ =\frac{1}{2}\left(\hat f(-N/2)e^{-iN/2x}+ \hat f(N/2 )e^{iN/2x}\right)+ \sum_{|m|>N/2}\hat f(m)e^{imx}, $$ where $N\in\mathbb Z$ has been fixed and $\ell$ varies in $\mathbb Z$. Somehow it makes sense since for every fixed $\ell\neq 0$, it holds $|k+\ell N|>N/2$. however I cannot conclude. Any help?