Evaluating $\sum_{n∈N}1/n^2$with the Poisson summation formula on Fejér kernel

Question

I'm asked to apply the Poisson summation formula to the Fejér kernel to obtain an expression for $$\sum_0^{\infty}1/(2k+1)^2$$. Hence show $$\sum_1^{\infty}1/n^2=\pi^2/6$$ Here Fejér kernel is given by $$F_n=\frac{1}{n}(\frac{\sin(\pi n x)}{\sin(\pi x)})^2$$ with period of 1.

I can only show that $$\hat{F_n}(\xi)=(1-\frac{|\xi|}{n})\chi_{[-n,n]}$$

Any hints would be appretiated.

Answer 1

There is a formula for Fejer kernel in terms of Dirichlet kernel $D_n$. It is easy to write down $\hat {D_n}$ so you can compute $\hat {F_n}$ easily from this. I will leave the starightforward calculations to you. I hope this hint helps.