Define $$\mathbf{F}_R(t) =
\begin{cases}
R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt]
R & t = 0
\end{cases}
$$
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.
Best Answer
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.