Let's say that we want to digitally sample an analog signal and have a sampling frequency of 8000 samples/second. Usually the maximum frequency that can be captured is the half of the sampling frequency; that is the maximum frequency that can be captured is approximately 4KHz for the example above. But what is the mathematical relationship between the two? Thank you.
The maximum frequency is also limited by how long you sample at a given sample rate. If you only sample for a finite length of time at 8000 sps, the bandwidth will be less than 4kHz. A 4000 Hz signal (or any integer multiple thereof) will be ambiguous, or simply disappear.
A signal somewhat below the Nyquist frequency (depending on how long you sample at a given S/N or accuracy) can be "captured", as well as any signals between two successive integer multiples of Fs/2 given a-priori knowledge of no aliasing from any other spectra inside any other folds.