Consider $$f(x) = \frac{1-\cos x}{\pi x^2}.$$
How can I prove the following?
$$G(t) = \int_\mathbb{R} e^{itx}f(x) dx = \begin{cases}1-|t|, &|t| \le 1 \\ 0, &|t|>1 \end{cases}$$ and $$G(t) = \frac{1}{2} + \frac{4}{\pi^2} \sum_{k=1}^\infty \frac{\cos((2k-1)t\pi)}{(2k-1)^2} \quad \forall t \in [-1,1]$$