Computing Fourier Integrals

Fast Fourier Transforms (or FFT's) are the first-stop for the numerical computation of Fourier transforms, but I recently learned that they are not well suited to evaluating the Fourier integral $latex \tilde{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{i\omega t} \,\textrm{d} t. $ This is stated explicitly in Numerical Recipes, and to borrow the words of a 1920's…