  Inference of transfer operators from data is often formulated as a classical
problem that hinges on the Ulam method. The conventional description, known as
the Ulam-Galerkin method, involves projecting onto basis functions represented
as characteristic functions supported over a fine grid of rectangles. From this
perspective, the Ulam-Galerkin approach can be interpreted as density
estimation using the histogram method. In this study, we recast the problem
within the framework of statistical density estimation. This alternative
perspective allows for an explicit and rigorous analysis of bias and variance,
thereby facilitating a discussion on the mean square error. Through
comprehensive examples utilizing the logistic map and a Markov map, we
demonstrate the validity and effectiveness of this approach in estimating the
eigenvectors of the Frobenius-Perron operator. We compare the performance of
Histogram Density Estimation(HDE) and Kernel Density Estimation(KDE) methods
and find that KDE generally outperforms HDE in terms of accuracy. However, it
is important to note that KDE exhibits limitations around boundary points and
jumps. Based on our research findings, we suggest the possibility of
incorporating other density estimation methods into this field and propose
future investigations into the application of KDE-based estimation for
high-dimensional maps. These findings provide valuable insights for researchers
and practitioners working on estimating the Frobenius-Perron operator and
highlight the potential of density estimation techniques in this area of study.
  Keywords: Transfer Operators; Frobenius-Perron operator; probability density
estimation; Ulam-Galerkin method; Kernel Density Estimation; Histogram Density
Estimation.
