A new numerical Fourier transform in d-dimensions

The classical method of numerically computing Fourier transforms of digitized functions in one or in d-dimensions is the so-called discrete Fourier transform (DFT) efficiently implemented as fast Fourier transform (FFT) algorithms. In many cases, the DFT is not an adequate approximation to the continuous Fourier transform, and because the DFT is periodical, spectrum aliasing may occur. The method presented in this contribution provides accurate approximations of the continuous Fourier transform with similar time complexity. The assumption of signal periodicity is no longer posed and allows the computation of numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT. In addition, this method yields accurate numerical derivatives of any order and polynomial splines of any odd degree. The numerical error on results is easily estimated. The method is developed in one and in d dimensions, and numerical examples are presented.