# Density function approximation with respect to $L^1$ distance

Given iid samples $$X_1,...,X_N$$ drawn from some unknown distribution with not necessarily continuous density function $$f(x)$$ are there any theorems/papers where based on the data $$X_1,...,X_N$$ an estimator $$f_N(x)$$ of $$f(x)$$ is defined and the approximation rate $$\int_{\mathbb{R}}|f(x)-f_N(x)|dx$$ is estimated?

The first systematic single-source examination of density estimates. It develops, from first principles, the natural'' theory for density estimation, L1, and shows why the classical L2 theory masks some fundamental properties of density estimates.