# [Physics] Duality and Fourier Transforms

dualityfourier transform

$(FF(f))(x)=2\pi f(-x)$, where $F$ is the Fourier transform

and $F(f(x-a))(k)=\exp(-ika) X(k)$ where $X(k)=F(f(x))$

implies $F(\exp(iax)f(x))(k)=X(k-a)$.

But I don't see how that is done… I am quite happy with getting $F^{-1}X(k-a)=\exp(iax)f(x)$ by brute force calculation. I would like to see how to use duality though.

You need to know the basic Fourier transform delta-function identity

$$\int_{-\infty}^{\infty} e^{ikx} {dk\over 2\pi} = \delta(x)$$

Which implies Fourier inversion. Proving this identity is slightly subtle, because the right hand side is a distribution, but you can do the integral explicitly over a long interval from -M to M to get an object which has a unit integral and is shrinking in size with M as 1/M, so it must be a delta-function in any reasonable sense of limits.

The double fourier-transform is

$$FF(f) (x') = \int dk e^{ix'k} (\int dk e^{ikx} f(x))$$

And you can do the k integral using the identity to get the result.