On density estimation via fourier series

In this paper, we consider the classical statistical problem of probability density estimation based on a sample from this distribution. This problem naturally arises in many applications when one aims at investigation of a probability structure in a random process. For instance, it is possible to identify some structure in a complex system using density estimation. In this paper, a new approach to estimate a density function is proposed. This approach is based on approximation of a log-density via Fourier series with coefficients obtained by solving a system of linear equations. Analysis of theoretical properties of such estimate is the main purpose of this work. As the main results, we prove bounds on the difference between target density and its approximation in the supremum norm and the Kullback-Leibler divergence. Obtained rates are parametric and have order with high probability, which is a standard rate in parametric estimation problems. The constants in the rates are obtained up to an absolute factor, which means that we investigated the dependence on all parameters. As a numerical example, we consider a problem of Cauchy density estimation.