You need both, square integrable and smooth. Both is provided by the condition band-limited. And no, you can't sample a sinusoid. That is, of course you can sample it, but the reconstruction formula will not converge. Since the reconstruction formula is essentially a representation in an orthonormal basis, it converges in the L2 sense exactly if the coefficient sequence is square summable, and the sample sequence of a sinusoid is not square summable because of its periodicity. But that was already discussed ad nauseatum years ago, so consult the archive pages to find the points of view of the participants.