I'd like to prove that the only possible solution to $\displaystyle f = f*f$ with $\displaystyle f \in L^1$ is f = 0. So, I have the following;

1. Take the fourier transform of both sides

$\displaystyle \hat{f} = \widehat{f*f} = \sqrt{2\pi}\hat{f}\hat{f}$

2. Cancelling we get

$\displaystyle \hat{f} = \frac{1}{\sqrt{2\pi}}$

3. Writing the convolution outright and cancelling appropriately, we obtain

$\displaystyle \int e^{-ix\xi} f(x) dx = 1$

In $\displaystyle L^1$, this must mean that f = 0, given that the indefinite integral is constant.

I'm unsure about this - any pointers would be much appreciated!