# Fourier Transforms: Proving operational properties

1. Oct 23, 2008

### Niles

1. The problem statement, all variables and given/known data
Hi all.

I wish to prove the following property of a Fourier transform:
$$F(f)(x) = F^{-1}(f)(-x),$$
which means that the Fourier transform of a function f in the x-variable is equal to the inverse Fourier transform in the -x-variable. This is proven here:

http://www.sunlightd.com/Fourier/Duality.aspx

Now I wish to prove the following:
$$F(f)(-x) = F^{-1}(f)(x),$$
but I cannot get started. I am not sure of what substitutions to make. Can you give me a hint?

sincerely,
Niles.

2. Oct 23, 2008

### CompuChip

If the first formula holds for any y, then just take x = -y. That's just it, isn't it?

3. Oct 23, 2008

### Niles

I guess you are right. Thanks :-)