Poisson summation formula clarification regarding Fejer kernel Define $$\mathbf{F}_R(t) = 
  \begin{cases}
    R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 &amp;  t \neq 0\\[10pt]
   R &amp;  t = 0
  \end{cases}
$$

A problem in Stein's Fourier Analysis asks us to prove that the periodization of $\mathbf{F}_R(t)$ is equal to the Fejer kernel on the circle

i.e.


  
    $$\sum_{n=-\infty}^{\infty}\mathbf{F}_N(x+n) = \sum_{n=-N}^{N}\left(1-\frac{|n|}{N}\right)e^{2 \pi i n x} = \frac{1}{N} \frac{\sin^2(N \pi x)}{\sin^2(\pi x)} $$ 
  


for $N \geq 1$ an integer

This strongly suggests an application of Poisson summation is needed, which would mean that we need to calculate $$\sum_{n=-\infty}^{\infty}\hat{\mathbf{F}}_N(n)e^{2 \pi i n x}$$

correct?

However, as I don't see how we can go from an infinite series to a finite series using Poisson, I assume we have to show that the above series converges to the closed form expression above?  I'm still a bit unclear as to how apply Poisson in this case: if this is correct, any hints as to how to tackle the integral
$$\int_{-\infty}^{\infty} \mathbf{F}_N(x)e^{-2 \pi i n x}dx$$ would be appreciated.
How about trying the following alternative approach. First, note that for integer $ N $, the numerator of $ F_N(t) $ factors out since:
$$
\sin^2(\pi N (t + n)) = \sin^2(\pi N t + \pi N n) = \sin^2(\pi N t)
$$
The remaining part is given by the identity:
$$
\sum_{n= -\infty}^{\infty} \frac{1}{\pi^2 (t+n)^2} = \frac{1}{\sin^2(\pi t)}
$$
I am not entirely sure how to prove this identity...but I hope this will help anyway.
Hint: You can make use of the function g in Exercise 2 of chapter 5 and the Fourier inversion formula to handle the inner integral you get as you apply the Poisson summation formula, i.e. the integral you are asking for.