Proof: Fourier Transform of f(x) = f(-x)

[Chen]

How can I prove that doing a Fourier transform on a function f(x) twice gives back f(-x)?

Thanks..

 

[Tide]

I think you're asking about the double integral

$$\int_{-\infty}^{\infty} e^{i kx} dk \int_{-\infty}^{\infty}e^{ikx'} f(x')dx'$$

If so, then the outer integral

$$\int_{-\infty}^{\infty}e^{i k (x+x')} dk = \delta (x+x')$$

i.e. the Dirac delta function and you arrive at your result upon evaluating the inner integral. (That's ignoring factors of $2\pi$ which I am sure you can handle!)

 

[Chen]

I should hope so. Thanks!