Timeline for Why is the Fourier transform so ubiquitous?
Current License: CC BY-SA 4.0
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| Mar 7, 2020 at 3:07 | comment | added | Michael Engelhardt | @DisplayName - What you describe sounds like what I would call an adiabatic approximation. Indeed, there are important systems for which this is applicable. It does require specific conditions to be fulfilled, as you note - so this is a bit different from saying, "Every system ..." | |
| Mar 6, 2020 at 21:50 | comment | added | Display Name | @MichaelEngelhardt Sticking around our favorite stationary point in time works if we are allowed to pick several stationary points in sequence and patch the local solutions together. When the excitation frequencies are much higher than the frequency of time varying this can even lead in the infinite limit to an important approximation technique that I forget the name of. | |
| Feb 19, 2020 at 4:15 | comment | added | Michael Engelhardt | @LSpice - Yes, that statement also struck me as a bit odd. Of course, there are plenty of systems that are neither time-invariant even for arbitrarily small times nor harmonic even for arbitrarily small amplitudes, so the "every system is" should be qualified in any case. But if we stick with the most common examples, which are indeed harmonic for small amplitudes, they usually are time-invariant altogether, with no "for small enough times" qualifier needed at all. And of course, as far as time evolution goes, we don't get to stick around at our favorite stationary point in time. | |
| Feb 19, 2020 at 0:35 | comment | added | LSpice | There's something disturbing about the statement "every system is time-invariant for small enough times" that somehow doesn't disturb me about "every system is linear for small enough amplitudes." | |
| Feb 14, 2020 at 16:06 | comment | added | Michael Engelhardt | @DisplayName - indeed, in physics, one often refers to "time shifts" as "translations in time" ... | |
| Feb 14, 2020 at 15:56 | comment | added | Display Name | To add to this answer, exponential functions are also eigenfunctions of the time shift operator. This means that any operator that commutes with the time shift is diagonalized in a basis of exponentials, or in other words that the Fourier (or Laplace, which includes exponential growth and decay) transform is appropriate for all linear, time invariant systems. Every system is linear for small enough amplitudes and time-invariant for small enough times, so there you go. | |
| Feb 13, 2020 at 2:49 | review | Suggested edits | |||
| Feb 13, 2020 at 3:00 | |||||
| Feb 11, 2020 at 20:58 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |