## Analytic expression of DFT of sinc sequence

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)?

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 The question of how to give a precise meaning to the Fourier series is very tricky and deep, and is only for experts in Analysis. I'd advise you to stay well clear of this topic if at all possible! Sorry, but also your question is rather poorly written; your $m$, $n$, $k$, $t$ in the equation all look very confused. – Zen Harper Mar 11 2011 at 7:07