The classical method of numerically computing the Fourier transform 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 of the continuous Fourier transform. 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 to compute numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT. In image processing this behavior is highly welcomed since it allows to obtain the Fourier transform of an image without the usual interferences of the periodicity of the classical DFT. The mathematical method is developed and numerical examples are presented.