Hi. According to my literature a "perfectly" reconstructed sampled signal can be represented as
$$ y{\left ( t \right )}=\sum_{k=-\infty }^{\infty }x\left ( kT \right ) sinc{\left ( \frac{\pi t}{T}-k\pi \right )} $$
Since the sinc factor is the only one that depends on t, I figured that the Fourier transform must be:
$$ Y{\left ( \omega \right )}=\sum_{k=-\infty }^{\infty }x\left ( kT \right )\pi rect{\left ( \frac{1}{2}\left ( \frac{\pi t}{T}-k\pi \right ) \right )} $$
But I can't see how these pi/2 spaced rectangles can reconstruct the signal in the frequency domain. What am I missing?