[Chinjut]

Consider a signal whose value at x seconds is f(2x) - 2 f(3x) + f(4x), where f(x) = sin(2πx)/x. Considering that the absolute frequencies of f(x) are uniformly distributed from 0 to 1 Hz, the absolute frequencies of this total signal should be constrained to between 2 and 4 Hz. Thus, a bandwidth of 2 Hz. But if we sample at 6 Hz (three times the bandwidth!) including x = 0, we'll get all zeros.

Granted, we might say that from the perspective of the complex Fourier transform using signed frequencies, the frequencies of this signal actually range over [-4 Hz, -2 Hz] U [+2 Hz, +4 Hz]. But I'm not sure that's the interpretation you had in mind.

Let me know if I've screwed anything up here!

[Chinjut]

That is, it's not quite as simple as saying you just need to sample at any frequency at least twice the bandwidth. Rather, it's the more complicated behavior described by this graph: https://en.wikipedia.org/wiki/Undersampling#/media/File:Samp.... That is, the general rule is that the ratio of the highest frequency in the signal to half the sample rate, and the ratio of the lowest frequency in the signal to half the sample rate, have to lie within an interval of consecutive natural numbers.

When the lowest frequency is zero, this is the familiar rule that the sample rate has to be at least twice the highest frequency in the signal. But more generally, it's more complicated.

[Chinjut]

Whoops, I should've pulled the division by x out of the definition of f. The example I had in mind was [sin(4πx) - 2 sin(6πx) + sin(8πx)]/x. [Another good example is [sin(6πx) - 2 sin(8πx) + sin(10πx)]/x, whose frequencies are between 3 Hz and 5 Hz, thus a bandwidth of 2 Hz, but sampling at 4 Hz or even 8Hz gets all zeroes.]

Anyway, the details on that example don't matter, the Wikipedia graph and article makes things more clear.