Bell Labs researcher Harry Nyquist develops Sampling Theory. It states provides that if a signal is sampled at twice its nominal highest frequency, the samples will contain all of the information in the original signal.
Which is true! Millions of mathematicians have prooved it.
if it was true, then you'd be able to sample a 22049Hz pure sine wave
at 44100Hz,
(you wouldn't need a filter, as there's no harmonics),
and then you'd be able to re-construct it.
Fourier is about finding the amplitude of harmonics within a periodic signal of
known frequency.
Now, the Nyquist Theorum refers to the possibility
of (for instance) sampling a 22049Hz pure sine at 44098Hz
with the phase aligned correctly.
In that case we get all the information.
(i.e. the amplitude, the one thing we didn't know already)
Proof only counts if it's relevant to the situation.
In the case of digital audio, that proof isn't relevant.
Which is clearly not true :-)
There's no way to keep the phase information for a signal sampled
at only twice it's frequency.
Only the amplitude.
guess what students ask their teachers of sampling theorems?
I have very little respect for educational establishments :-)
but I did actually stay long enough at college to see fourier explained
They usually ask the same as you do and they get an answer they can understand. You have to do the mathematics. I do not know anybody who does the mathematics behind it, still claims that its not possible.
What usually is forgotten, is that the Nyquist theorem is aimed at infinite observation time.
it's "aimed" at a periodic signal of known period,
( so it's assumed infinite)
andy