The characteristic function of a probability distribuition is given by the Fourier transform of the distribuition, as follows: $$\phi(t)=\langle e^{itx} \rangle =\int_{-\infty}^{+\infty} \rho\left(x\right) e^{itx} dx $$
But is this transformation injective? Can two different functions $\rho(x)$ have the same characteristic function?