A filter with a narrower-than-Fourier Transform passband?

[virtualetters]

If I have a signal that I'd like to filter very precisely (say, a few Hz bandwidth), I feel like the "best" one can possibly do is to take a time sample long enough to get a discrete transform frequency resolution of the same width, erase all but the frequency component of interest, and then transform back.

But am I supposing the Fourier transform is more fundamental than it really is? Could I use, say, a recursive filter or other set of basis functions, to have a few Hz passband over timescales much shorter than the transform-limit? Or is the (Fourier) transform limit really a fundamental limit?

Thanks!

 

[mfb]

If your sample is shorter then it will necessarily have a larger width, and you won't be able to distinguish signals of similar frequency reliably. If you don't need that: Good.
Determining e.g. the central frequency works with much shorter samples. If you know your signal has a very narrow width then you can determine its frequency very accurately, not limited by the Fourier transformation.