The Fourier transform and the discrete Fourier transform

The authors establish sharp error bounds for the error committed in computing N values of the Fourier transform of a square summable function by means of the discrete Fourier transform. No assumptions are made as to the bandwidth or desired frequency resolution. Instead their error bound depends explicitly on five parameters: the number N of sampling points, the interval T where the samples are taken, the interval Omega where the Fourier transform is being approximated, and two extra parameters that account for local averaging in the time and frequency domains respectively. The resulting error bound is independent of the function in question and they find that, for an appropriate model of sampling, the error bound is minimised as a function of T, Omega by the Nyquist-Shannon choice 4T Omega =2N+1. They finally consider how these bounds are affected by some a priori knowledge about the class of functions under discussion. This is done in detail for band-limited functions.