Let us say that we have some Schwartz function $f : \mathbb{R} \to \mathbb{C}$. The Fourier transform of $f$ is given by: $$ \hat{f} (\xi ) =\frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} f(x)e^{-2\pi i x \xi } \mathrm{d}x $$ In what sense does the discrete Fourier transform, defined by:
$$ \hat{x}(k) = \sum_{j = 1}^{N - 1}x(j)e^{\frac{-2\pi i}{N} kj} $$
approximate the Fourier transform of some function? Note that I am not really looking for a rigorous explanation or something about modes of convergence, rather, why intuitively (say through the use of Riemann sums) the above can serve as a numerical approximation of a standard Fourier transform.