Let ${x_n}$ be an infinite sequence of samples from $x(t)=\text{sinc}(t)$ (assume samples are taken every $k$th sample). Now, we know that the Fourier transform of $x(t)$ is the top-hat function and vice-versa. However, I've read that to be able to write the discrete Fourier transform as
$X_m=\sum_{n=-\infty}^\infty x_k e^{imt}$
requires the $x_k$ to be absolutely summable. Is that correct, or am I wrong? Taking the DFT using matlab/mathematica certainly results in the top-hat function. But is writing it this way analytically correct (for this sequence)?
