abridged] A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of FFTs, which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value N=3, the number of FFTs required is 20. We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive explicit analytic expression for the spectrum, which allows us to quantify--and correct for--the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample. The analytical calculations are successfully checked against a cosmological N-body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier-Taylor estimator behaves well. We also show how to reach arbitrarily large dynamic range in Fourier space (i.e., high wavenumber), while keeping statistical errors in control, by appropriately folding the particle distribution.
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