Quantitative estimates: How well does the discrete Fourier transform approximate the Fourier transform on $\mathbb{R}$

In order to compute the Fourier transform of a function $f$ on the real line numerically, one samples $f$ on a grid and then takes the discrete Fourier transform. We derive exact error estimates for this procedure in terms of the decay and smoothness of $f$. The analysis provides a new recipe of how to relate the number of samples, the sampling interval, and the grid size.