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Then $ x(t)$ can be exactly reconstructed from its samples $ x_d(n)$ if $ X(j\omega)=0$ for all $ \vert\omega\vert\geq\pi/T$.D.3
Proof: From the continuous-time aliasing theorem (§D.2), we have that the discrete-time spectrum $ X_d(e^{j\theta})$ can be written in terms of the continuous-time spectrum $ X(j\omega)$ as
$\displaystyle X_d(e^{j\omega_d T}) = \frac \sum_{m=-\infty}^\infty X[j(\omega_d +m\Omega_s )] $
where $ \omega_d \in(-\pi/T,\pi/T)$ is the ``digital frequency'' variable. If $ X(j\omega)=0$ for all $ \vert\omega\vert\geq\Omega_s /2$, then the above infinite sum reduces to one term, the $ m=0$ term, and we have
$\displaystyle X_d(e^{j\omega_d T}) = \frac X(j\omega_d ), \quad \omega_d \in\left(-\frac{\pi},\frac{\pi}\right) $
At this point, we can see that the spectrum of the sampled signal $ x(nT)$ coincides with the nonzero spectrum of the continuous-time signal $ x(t)$...
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