Please help me prove this!
f(x) is piecewise continuous on any interval, and $$\int_{-\infty }^{\infty } |f(x)|dx < \infty$$
Here
$$f(x)=\int_{-\infty}^{\infty}[A(\alpha)\cos \alpha x + B (\alpha)\sin \alpha x]d \alpha$$
where
$$A(\alpha) = \frac{1}{\pi}\int_{-\infty}^{\infty}f(x) \cos \alpha x dx$$
and
$$B(\alpha) = \frac{1}{\pi}\int_{-\infty}^{\infty}f(x) \sin \alpha x dx.$$
How can I show formally that
$$\frac{1}{\pi}\int_{-\infty }^{\infty } f^2(x)dx = \int_{-\infty}^{\infty}[A^2 (\alpha) + B^2 (\alpha)]d \alpha.$$
I know that each of these are a Parseval's equality for the corresponding transforms.