Oh man this is one of my favorite topics.
So I think what this discussion is ultimately getting at is are all waveforms in physical reality actually composed of sine waves of various harmonics all superimposed on each other, or is that just a mathematical artifact of how the Fourier transform is formulated?
Remember, if you look at the spectrum of a waveform, what you’re going to see is the FFT of that waveform. So naturally one will see that the frequency composition of that wave will agree with what a Fourier transform of it says it will be because you’re doing a Fourier transform.
In other words, it shows that Fourier transforms agree with themselves. A bit circular, isn’t it?
However, I am here to tell you that not only is this fundamental to all waves, it underpins reality as we know it.
Quantum mechanics (which is really just waves doing wave stuff. It should be called quantum wave dynamics), for example, has quantized waveforms called wavefunctions. A wave function can be a single particle or an ensemble of them depending.
But this wave function is a complex valued function, with a real and imaginary component that are identical but out of phase by a constant phase shift. The time axis is space/position/distance in this case, however.
If you square the amplitude of both components and add them together, you get the probability of interacting with the thing represented by this wave function. And the phase shift is such that this number is always between 0 (the imaginary part yields a negative number when squared which cancels out the positive probability) and 1, the only valid range for a probability.
Position and energy/momentum are inversely related by the uncertainty principle. The more precisely you localize a particle, the wider the range of energies it can have. And vise versa.
The reason why is that physical reality, at the most fundamental level, physically manifests what one might think was just a mathematical formalism of the Fourier transform.
Imagine a sinewave that you want to turn into a wave packet, one that maybe had a single sharp peak in the center and falls off quickly, like the sinc function.
How would you do that?
Well, you could add harmonics. You can keep adding harmonics and localize this wave into a packet more more tightly, but at the cost of it no longer having a well-defined frequency. Now it exists as a combination of many frequencies superimposed.
Frequency is the same as energy/momentum for waves.
So when you have, say, an electron with a single possible energy level/frequency, it’s wave function is delocalized. It is spread out, it doesn’t exist in any specific location anymore than a sine wave exists at a given location in time.
If you want to localize it, this means injecting energy into it so it can be at multiple energy levels/frequencies. As you increase the uncertainty (add harmonics) of its energy/momentum, you can localize it by making alternate versions of itself where it has different energy levels/frequencies interfere with themselves.
In case the significance isn’t clear, this means that the very concept of location/position is merely an emergent property of the more fundamental wave dynamics that govern our physical reality.
So the Fourier transform has a physical manifestation and it is one that underpins physical reality itself. It is safe to say that it is something deeply fundamental to all continuous superimposable functions (which includes all waves of any basis), and that this is a real, physical thing and not just a result of how the math works.
FYI, this is how flash memory works. A fully isolated MOSFET gate has some charge injected into it via tunnel injection. This works by tightly confining the energy (frequency) of some electrons so they delocalize and one of the possible places they might exist is inside that isolated gate. Once this confinement is removed, some of them will be stuck on that gate.
You just wrote a bit by exploiting the physical manifestation of the Fourier transform. Kind of wild, isn’t it? 
Quick tangent: remember how I mentioned quantum wave functions are identical real and imaginary waves but locked in phase?
If this wave interacts with stuff over position, this will cause different interactions with the real and imaginary components and they will no longer be identical. This would permit negative probabilities to occur, which is no bueno.
We can add something called the lagrangian, it is a variable we just made up and added to the wave function’s equation. It is defined as a variable that has a value that varies with space/distance such that it ensures the real and complex parts of the wave function remain identical but out of phase even when this phase shift would result in different propagation.
If something that has a value at every location and changes over position sounds like a field, that’s because it is.
That variable we added to keep the real and imaginary components identical turns out to be a field we all hold near and dear:
It is the electromagnetic field.
I know no one asked but it’s so rare I can weasel it into anything and it’s just so cool that I couldn’t help myself.
