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.

翻译：从数据中推断传递算子通常被表述为一个依赖于Ulam方法的经典问题。传统描述称为Ulam-Galerkin方法，它涉及投影到由支撑在精细矩形网格上的特征函数表示的基函数上。从这个角度来看，Ulam-Galerkin方法可被解释为使用直方图方法进行密度估计。在本研究中，我们将该问题重新置于统计密度估计的框架内。这一替代视角允许对偏差和方差进行明确且严谨的分析，从而促进对均方误差的讨论。通过利用逻辑斯蒂映射和马尔可夫映射的综合示例，我们验证了该方法在估计Frobenius-Perron算子特征向量时的有效性和实用性。我们比较了直方图密度估计（HDE）和核密度估计（KDE）方法的性能，发现KDE通常在精度上优于HDE。然而，值得注意的是，KDE在边界点和跳跃点附近表现出局限性。基于我们的研究结果，我们建议将其他密度估计方法引入该领域的可能性，并提出未来对基于KDE的高维映射估计应用的研究方向。这些发现为从事Frobenius-Perron算子估计的研究人员和实践者提供了宝贵见解，并突显了密度估计技术在该研究领域的潜力。关键词：传递算子；Frobenius-Perron算子；概率密度估计；Ulam-Galerkin方法；核密度估计；直方图密度估计。