Sampling

Sample above twice the highest frequency or components fold back. An aliased tone appears at the distance to the nearest multiple of $f_s$.

Sampling maps the analogue frequency $\Omega$ to the digital frequency $\omega=\Omega T$, so a digital filter acting at $\omega$ behaves on the original signal as one at $\omega/T$. Decimation and interpolation each need a filter, and each for a different reason: one to prevent aliasing, the other to remove images.

Ideal reconstruction is a brick-wall low-pass filter, which in the time domain interpolates with sincs. Real converters use a zero-order hold instead, and that costs a predictable $\mathrm{sinc}$ droop across the band that has to be compensated.

When a tone is undersampled it does not vanish — it reappears somewhere inside the band, and the question is almost always where. Fold the frequency about multiples of $f_s$ until it lands in $[0,f_s/2]$.