On an application of a generalization of the discrete Fourier transform to short time series

A generalization of the discrete Fourier transform (DFT) is discussed. This generalization or GDFT provides a smooth interpolation between the points of the DFT. The GDFT of a sinusoidal function in a finite time window is (a) described in detail and (b) shown to coincide (aside from a simple scaling constant) with the corresponding Fourier transform, provided that certain conditions are satisfied by the sinusoidal parameters. The sinusoidal GDFT is proposed as a tool to investigate, (independently of any Fourier transform connection) the sinusoidal nature of time series. The method is applied successfully to the case of a specific trajectory of the Hénon and Heiles model.