This was triggered by the comments in this question.
I'm using this definition of the Shannon-Nyquist theorem (form wikipedia):
If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
I was under the impression that the Nyquist theorem is theoretically true in the following sense: if you decompose a signal into sinusoids, and then sample at 2x the highest frequency sinusoid, you can perfectly reproduce the original signal. This is because there's only one sinusoidal curve that fits all the samples at or below 1/2 the sampling rate, and if we're considering the highest frequency component in the original signal, it must be a sinusoid (or else it wouldn't be the highest frequency in the signal).
But others commenting on the question linked above said this only applies to continuous signals. One person elaborated on this as follows:
think about continuous signal reconstruction from discrete-time values: in theory, every value between two sample instants requires the sinc sidelobes of all countably infinitely many samples before and after. That's a bit problematic, especially because we don't know the future. Assuming periodic repetition is one of the common tricks to get around that, so that with only a limited amount of past and no future, we can reasonably sinc-interpolate
I don't really understand this - does it mean that what I said above about the Nyquist theorem is not completely true?