Let $\rho$ be a static, discrete or continuous charge distribution, and $\phi(\mathbf{r})$ the corresponding electric potential. We may expand $\phi$ in a multipole series, $$ \phi(\mathbf{r}) = \frac{1}{r}\sum_{n \geq 0} \int \text{d}^3r' \rho(\mathbf{r}')\left(\frac{r'}{r}\right)^nP_n(\cos\theta).$$ My question is, if given the multipole expansion, one can determine the source $\rho$.
Evidently, this is possible if given the multipole expansion of two equally and oppositely charged point sources a distance $a$ apart. From the dipole moment, one can deduce $a$, and thus reconstruct the charge distribution. But is this true in general?