Can anyone tell me how to prove the equation below using Fourier Transform for even function? I tried but I really have no idea.
$$\frac2\pi\int_0^\infty \frac{\sin\pi u \cos xu}u du = \begin{cases} 1,& (|x|\le\pi)\\ 0,& (|x|>\pi) \end{cases}$$
For Fourier Transform of an even function, what I was taught is as below:
If $f(x)$ is an even function the Fourier transform and the Inverse Fourier Transform is:
$F(\omega) = 2\int_0^\infty f(x)\cos \omega x dx$
$f(x) = \frac1\pi\int_0^\infty F(\omega)\cos \omega x d\omega$
Here is my though:
The left side is the Fourier transform of $f(u) = \frac{\sin\pi u}{\pi u}$
And I stuck here.