Find the Fourier transform the function

Question

The given function is $$f(x) = ({a - |x| , |x| < a})$$ and hence deduce that $$\int_{0}^{\infty} \left(\frac{\sin(2t)}{t}\right)^2 dt = \frac{\pi}{2}.$$ Can anyone please solve this?

Answer 1

Note that $f(x)=\max\{0,a-|x|\}$. According to the definition of Fourier transform, you have to evaluate $$\hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)e^{-ix\xi}dx=\int_{-a}^a(a-|x|)e^{-ix\xi}dx.$$ Then, in order to deduce the integral, $$\int_{0}^{\infty} \left(\frac{\sin(2t)}{t}\right)^2 dt= \frac{1}{2}\int_{-\infty}^{\infty} \left(\frac{\sin(2t)}{t}\right)^2 dt= \int_{-\infty}^{\infty} \left(\frac{\sin(s)}{s}\right)^2 ds$$ consider the inverse Fourier transform: $$\int_{-\infty}^{\infty}\hat{f}(\xi)e^{ix\xi}d\xi=2\pi f(x).$$ Can you take it from here?