In Fourier analysis it is said that the only way to have 0 uncertainty in a monochromatic wave's frequency is if you FT it over an infinite domain (otherwise it does not produce a delta function).
This makes sense to me as cosine (etc) waves with slight frequency differences can be negligibly different unless you compare them over many many cycles. However it seems to me they would still only be 'negligibly different' and not completely the same. In theory if we measured every point in a wave with 0 uncertainty, can't we just compare which single frequency passes through all of those points with absolutely 0 error to get the frequency perfectly without using an infinite domain?
If you only examine a finite segment of your "apparently a monochromatic cosine", how do you know that it continues to be the same cosine outside that finite segment? Only if you inspect the entire curve can you know that it really is the cosine that you inferred from the finite segment.
Are you familiar with frequency modulation? This is a real practice, used for FM radio that produces a time varying frequency at fixed amplitude. Just because an FM signal has a fixed frequency for a finite time interval does not mean it has that frequency for all times.
Now if, miraculously, you know you had a monochromatic cosine wave (i.e., you know something about the wave at all times outside your finite sampling window), then the noiseless wave data in your finite window are sufficient to determine exactly what frequency that wave has. This requires knowing more than just what is in the window.