oc.optimization control - Approximating a probability distribution by a mixture

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!