The question below is really bugging me ...Given that the following function,
$$f(x) = \Theta(x + 1) - \Theta(x - 1)\,,$$
(where $\Theta(x)$ is the Heaviside step function) has a Fourier transform:
$$\widetilde{f}(k) = 2\frac{\sin k}{k}$$
where in this case we define the Fourier transform as
$$\widetilde{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx}\,dx\,,$$
evaluate the following integral:
$$\int_{-\infty}^{\infty}\frac{\sin^2x}{x^2}\,dx\,.$$
So far, I've considered that the Fourier transform of $c(x) = f(x) \ast g(x)$ (with '$\ast$' denoting convolution) is equal to $\widetilde{c}(k) = \widetilde{f}(k) \widetilde{g}(k)$ by the convolution theorem, hence if $g(x) = f(x)$, this gives a starting point. I have so far failed to manipulate any integrals into the correct form, though!
I feel like the solution is much simpler than I am imagining it to be ...