Let us consider a probability distribution $g_n$ for $n \in \mathbb{N}$ which we want to approximate by a mixture of $f_n(\lambda)$ where $\lambda \in \mathbb{R}$ is a parameter.

Are there known techniques that allow one to find the mixture minimizing the $L^1$ norm:
\begin{equation}
\min_{p \geq 0} \sum_{n=0}^{\infty} \left|g_n - \int \rm{d} \lambda \;  p(\lambda) f_n(\lambda) \right| \; \; ?
\end{equation} 

Any pointer to the relevant literature would be greatly appreciated.
Thanks a lot!