Efficiently implementing nonlinear Bayesian estimators is still not a fully solved problem. For practical applications, a trade-off between estimation quality and demand on computational resources has to be found. In this paper, the use of nonnegative Fourier series, so-called Fourier densities, for Bayesian estimation is proposed. By using the absolute square of Fourier series for the density representation, it is ensured that the density stays nonnegative. Nonetheless, approximation of arbitrary probability density functions can be made by using the Fourier integral formula. An efficient bayesian estimator algorithm with constant complexity for nonnegative Fourier series is derived and demonstrated by means of an example.