Let us consider a probability distribution $(g_n)_{n \in \mathbb{N}}$ which we want to approximate by a mixture of $(f_n(\lambda))_{n \in \mathbb{N}}$ 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} \sum_{n=0}^{\infty} \left|g_n - \int \rm{d} \lambda \; p(\lambda) f_n(\lambda) \right| \end{equation} where $p(\lambda)$ is a normalized probability distribution?

The motivation of this problem is linked to experimental physics: ideally one would like to generate an experimental process characterized by the probability distribution $g$ but this is really not practical. What is really easy, however, is to generate an experimental process with the distribution $f(\lambda)$ where $\lambda$ is a tunable parameter. Therefore, the goal is to approximate $g$ as closely as possible with such a mixture of $f(\lambda)$, where the distance between the two distribution is computed with the $L^1$ norm, that is, I want to minimize the variation distance between the two distributions.

In the specific problem I consider, $f(\lambda)$ is a Poisson distribution with parameter $\lambda \geq 0$, but I really am interested in a general method to approach this problem-

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