Frequency translation of an ideal low pass filter
Multiplying the impulse response by $e^{jTn}$ would give a complex impulse response and a complex-valued filter output. Seems that they were aiming for a real-valued output. They did this by summing two filters, one with the pass band at $T - W \le \omega \le T + W$ and the other with the pass band at $-T - W \le \omega \le -T + W$. If there is overlap between the two pass bands then the frequency response of the composite filter will have a weird shape like:



Seems that they did not intend there to be overlap, but that $T \gt W$.