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.

翻译：为了数值计算实线上函数$f$的傅里叶变换，需在网格上对$f$进行采样，然后进行离散傅里叶变换。我们基于$f$的衰减性与光滑性推导了该过程的精确误差估计。该分析提供了一种关联采样点数、采样间隔与网格尺寸的新方法。