Hello!

Im not in a hurry, but I dont know where to post this (easy) question.

I know this:

$\displaystyle |\hat{f}(\xi)|^2 := |\int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx, |^2 < \infty$

$\displaystyle |\hat{g}(\xi)|^2 := |\int_{-\infty}^{\infty} g(x)\ e^{- 2\pi i x \xi}\,dx, |^2 < \infty$

(Fourier transform)

Now I want to show that

$\displaystyle |\hat{f}(\xi)|^2 + |\hat{g}(\xi)|^2 < \infty$

I guess this is true, because I think it is obvious, but isn't there a professional proof on this?

What do you think?

Any comments would be much appreciated.

Best regards,

Rapha