What Is the Fourier Transform?

I have a pet theory that the reason why the FT, and other transforms (generating functions, Mellin/Laplace/Legendre/Haar), are so useful is because many real world functions are sparse and lend themselves to compressed sensing.

The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the problem. Why? Pet theory: because many functions that look complex are actually composed of simpler building in the transformed space.

Take the sound wave of a fly and it looks horribly complex. Pump it through the FT and you find a main driver of the wings beating at a single frequency. Take the sum of two sine waves and it looks a mess. Take the FT and you see the signal neatly broken into two peaks. Etc.

The use of the FT (or DCT or whatever) for JPEG, MP3 or the like, is basically exploiting this fact by noticing the signal response for human hearing and seeing it's not uniform, and so can be "compressed" by throwing away frequencies we don't care about.

The "magic" of the FT, and other transforms, isn't so much that it transforms the signal into a set of orthogonal basis but that many signals we care about are actually formed from a small set of these signals, allowing the FT and cousins to notice and separate them out more easily.

As mentioned by other commenters, a reason for the FT's dominance in particular is because sine, cosine, and complex exponentials are the eigenfunctions of the derivative operator. Since so many real-world systems are governed by differential equations, the Fourier Transform becomes a natural lens to analyze these systems. Sound waves are one (of many) examples.

And there's another good reason why so many real-world signals are sparse (as you say) in the FT domain in particular: because so many real-world systems involve periodic motion (rotating motors, fly's wings as you noted, etc). When the system is periodic, the FT will compress the signals very effectively because every signal has to be harmonic of the fundamental frequency.

As you noted, it’s about what’s important to us. The physical function may or may not be sparse, but our brain model is guaranteed to be sparse. A note played on a violin is anything but a sine function, yet our brains associate it with a single idealized tone. Our world model is super compressed.

The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion".

Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.

> The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion". > > Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.

I would say that the it's very difficult to imagine a world that would not be governed by differential equations. So it's not just that life wouldn't exist it's that there wouldn't be anything like the laws of physics.

As a side note chaotic systems are often better analysed in the FT domain, so even in a world of chaotic systems (and there are many in our world, and I'd argue that if there wasn't life would not exist either) the FT remains a powerful tool

That first question is a tautology. It’s like asking “Why is a screwdriver so perfect for turning screws?”

We have discovered a method (calculus) to mathematcally describe continuous functions of various sorts and within calculus there is a particular toolbox (differential and partial differential equations) we have built to mathematically describe systems that are changing by describing that change.

The fact that systems which change are well-described by the thing we have made to describe systems which change shouldn’t be at all surprising. We have been working on this since the 18th century and Euler and many other of the smartest humans ever devoted considerable effort to making it this good.

When you look at things like the chaotic behaviour of a double pendulum, you see how the real world is extremely difficult to capture precisely and as good as our system is, it still has shortcomings even in very simple cases.

What ought to be surprising is that the "thing" itself doesn't change.

A learning that describes chaos well enough may not want to be associated with "calculus", or even "math" (ask a friendly reverse mathematician about that)

https://www.johndcook.com/blog/2021/04/09/period-three-impli...

Somewhat tangentially, if Ptolemy I had responded (to Euclid) with anything less specific ---but much more personal--- than "check your postulate", we wouldn't have had to wait one millennium.

(Fermat did the best he could given margin & ego, so that took only a century or so (for that country to come up with a workable strategy))

Less tangentially, I'd generalize Quigley by mentioning that groups of hominids stymie themselves with a kind of emergent narcissism. After all, heuristics,rules and even values informed by experience & intuition are a sort of arrogance. "Tautology" should be outlawed in favour of "Narcissism" as a prosocial gaslighting term :)

As an aside, here's a relevant video about the (sometimes not) chaotic nature of double pendulums: https://www.youtube.com/watch?v=dtjb2OhEQcU

Differential equations aren't limited to describing stable systems, though, and there are chaotic systems that are also in some sense stable.

Ordinary differential equations can describe any system with a finite number of state variables that change continuously (as opposed to instantaneously jumping from one state to another without going through states in between) and as a function of the system's current state (as opposed to nondeterministically or under the influence of the past or future or some kind of supernatural entity).

Partial differential equations extend this to systems with infinite numbers of variables as long as the variables are organized in the form of continuous "fields" whose behavior is locally determined in a certain sense—things like the temperature that Fourier was investigating, which has an infinite number of different values along the length of an iron rod, or density, or pressure, or voltage.

It turns out that a pretty large fraction of the phenomena we experience do behave this way. It might be tempting to claim that it's obvious that the universe works this way, but that's only because you've grown up with the idea and never seriously questioned it. Consider that it isn't obvious to anyone who believes in an afterlife, or to Stephen Wolfram (who thinks continuity may be an illusion), or to anyone who bets on the lottery or believes in astrology.

But it is at least an excellent approximation that covers all phenomena that can be predicted by classical physics and most of quantum mechanics as well.

As a result, the Fourier and Laplace transforms are extremely broadly applicable, at least with respect to the physical world. In an engineering curriculum, the class that focuses most intensively on these applications is usually given the grandiose title "Signals and Systems".

> Well, stable systems are can either be stationary or oscillatory.

In practice this is probably true, but I can see another possibility. The system could follow a trajectory that bounces around endlessly in some box without ever repeating or escaping the box.

Natura non facit saltus.

https://en.wikipedia.org/wiki/Natura_non_facit_saltus

One amazing application of spectral theory I always harp on when this topic comes up is Chebfun[1]. Trefethen's Spectral Methods in Matlab is also wonderful.

[1] http://www.chebfun.org/

On the simplest end of that spectrum, Taylor series are useful because many real-world dynamics can be approximated as a "primarily linear behavior" + "nonlinear effects."

(And cases where that isn't true can still be instructive - a Taylor series expansion for air resistance gives a linear term representing the viscosity of the air and a quadratic term representing displacement of volumes of air. For ordinary air the linear component will have a small coefficient compared to the quadratic component.)

I haven't read it! Thanks for the recommendation!

You can treat that, and scientist often do treat it, as a stationary system with some error bounds.

For example, the concept of homeostasis in biology is like this. Lots of things are happening inside the living body, but it's still effectively at a functional equilibrium.

Similarly, lots of dynamic things are happening inside the Sun (or any star), but from the perspective of Earth, it is more or less stationary, because the behavior of the sun won't escape some bounds for billions of years.

I agree broadly with what you say. I didn't have time to make a more comprehensive comment.

If this box was of a bounded size then that trajectory would have interesting property - there are chunks of time you can edit out such that what remains will look as if they are converging on a point.

I suspect you will find ergodicity interesting.

Apparently he uploaded it to ResearchGate: https://www.researchgate.net/profile/Hector-Carmenate/post/H...