Is the frequency domain a real place?
lcamtuf.substack.com
lcamtuf.substack.com
The interesting thing about time-dependent signals (or any "pretty" function, really) is that they live in an infinite-dimensional vector space, which is hard to imagine; but (besides some important technicalities) the math works mostly the same way: signals as infinite-dimensional vectors can be represented in a lot of bases. One representation is the Fourier transform, where the basis vectors are harmonic functions. The "map" showing the shape of a signal as a combination of infinitely many harmonic functions -- i.e. the frequency domain -- is just as real as any other map with different basis vectors, e.g. the Walsh–Hadamard transform mentioned in the article. And, crucially, the original time-domain representation is also just one map showing us the signal, though it is often the most natural to us.
"Mathematically, the Fourier transform is "simply" a way of representing time signals in a certain orthogonal vectorial basis."
Not just time signals but any piecewise continuous and differentiable as well as Dirichlet integrable function. This has many applications, just a few examples from the top of my head: image processing, solving differential equations, fast multiplication.
I'd also like to add that from a mathematical point of view these transforms are "lossless" in the sense that the transformed function has the exact same information as the original and you can get back the exact original even if all you have is the transform.
I feel this often gets lost when people approach the Fourier transform from a more engineering perspective, not at least because we often do the transform to throw away unwanted information, like certain frequency components.
In the end it really is just one of many perspectives to look at a function.
That was my problem as well. My first introduction to Fourier transforms was through more of an engineering lens. I remember having trouble with the _inverse_ Fourier transform. I was OK with a Fourier inverse of an already transformed function but I wasn't quite sure what that would mean when applied to a non-transformed, "regular" function.
Operations on cepstra are labelled quefrency analysis (or quefrency alanysis[1]), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with kepstrum.
Have you gained some intuition/understanding for this?
I tried a few inputs in WolframAlpha, but unless I manually type in the integral for the inverse transform there's not even a graph :) (and I have no idea whether it's even the same thing without putting a `t` in the exponent and wrapping it in an f(t) = ... )
https://www.wolframalpha.com/input?i=integral+%28sin%28x%29+...
Every continuous periodic function turns into a discrete aperiodic one when transformed. Works both ways.
Continuous aperiodic stays continuous aperiodic. Discrete periodic stays discrete periodic.
If you apply fourier transform 4 times you get your original function back. You can think of it as 90 degree rotation. Inverse transform just rotates it in the opposite direction.
The rotation analog is not even too far fetched as fractional fourier transform allows you to do an arbitrary angle rotation.
F0: original signal
F1: frequency domain signal
F2: reverse time signal
F3: inverse fourier signal
F4: original signal
Also, has further weird applications I've never heard of with "Fractional Fourier Transforms" [1] which can apparently result in smooth smears of time -> frequency domain [2].
[1] https://en.wikipedia.org/wiki/Fractional_Fourier_transform
[2] https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stevencys.j...
The Fractional DFT part though, doesn't seem to do anything no matter the function chosen. Firefox 125.
Edit: Nvm, figured it out. Have to visualize from the top down to see the Fractional DFT portion. Haven't seen many visualization systems where each orientation shows a different type of data. Actually a pretty neat idea from a UI perspective.
This means that sure, the Fourier transform itself isn't lossy (garbage in, garbage out) but Fourier transforms would be used in contexts where loss are introduced. If I have a real perfect square wave, and I want to a take a fourier transform of it, the sampling is going to introduce loss, so to associate sampling losses with the transform itself is fair. Real square wave ran through a DFT program on my computer is going to spit out an approximation of a square wave -- loss.
But also, the question this is all responding to was effectively "why would engineers associate Fourier transforms with loss" and the answer is simply "because the techniques used in calculating most Fourier transforms are going to inherently put a frequency limit and anything beyond that will be lost or show up as an artifact". Engineers work with real world constraints and tend to be hyper aware of those constraints even if they often don't matter.
The Fourier transform of a discrete signal repeats in the frequency domain. For example, [1, -1, 1] could be a sine wave with the exact half of the sampling frequency going from 1 to -1 back to 1 once .... Or it could be a sine wave with double the sampling frequency that is actually going from 1 to -1 to 1 to -1 all within the gap of the first 2 samples. Or it could be 3x the sampling frequency, 4x the sampling frequency, etc. The solution is to only keep the part of the transform that is below the Nyquist limit, because we don't have a sampling rate accurate enough to measure the higher frequencies, so just assuming they dont exist. This also means that if the source signal WAS in fact 4x the sampling frequency, we will see a spike at 1/2 the sampling frequency in the fourier transform, and when we re-create the signal, it will be completely wrong.
So unless you have analog hardware for measuring the Fourier transform (or are working purely in a non-physical mathematical domain, like "i have a sine wave" which can be perfectly represented), you are naturally going to be taking discrete samples of a signal to measure the Fourier transform, which means you are going to be losing any part of the signal that doesn't adhere to sampling rates.
Because my engineering courses were so heavily focused on digital signal processing, when I hear "fourier transform" i immediately think of "discrete fourier transform" and loss is immediately applicable.
Remember that it is an integral transform. Basically, any data on a set of vanishing measure can be lost or corrupted. Unfortunately it can even be the case the deviation around some points is unbounded.
And there's no requirement that they be perpendicular is there? The second just needs 'some amount of perpendicular', North and North-East for example? Since any [n, e] can also be described as [(1-sqrt(2)*e)*n, sqrt(2)*e] in the latter. (I think that's right, but my main point is you can do it, not the particular value there, and if that's way off I'll blame the fever.)
In particular, for multidimensional spaces, the usual multidimensional Fourier transform only really works if you have a flat metric on that space (I.e. no curvature). That’s a bit of a warning signal given that our universe itself is curved.
There was some very interesting work recently where it was shown how to generalize Fourier series to certain hyperbolic lattices [1], and one important outcome of that work is that the analog of the Fourier space is actually higher dimensional than the position space.
Furthermore, the dimensionality of the ‘Fourier space’ in this case depends on the lattice discretization. One 2D lattice discretization may have a 4D frequency-like domain, and another 2D lattice might have a 8D frequency-like domain.
[1] https://arxiv.org/abs/2108.09314 or https://www.pnas.org/doi/full/10.1073/pnas.2116869119
What does this has to do with whether they are a different basis for cases where we don't account for curvature? This seems completely irrelevant, sure the tool can't be used in some cases but it can be used as a basis change in other cases.
[1] https://www-users.cse.umn.edu/~garrett/m/repns/notes_2014-15... (linear algebra required at least to the degree that one is comfortable with the difference between a matrix and an operator and knows what a direct sum is)
In uniformly curved 2D hyperbolic spaces, it turns out that there is a higher dimensional non-Abelian Fuchsian translation group defined on a higher genus torus.
> In particular, for multidimensional spaces, the usual multidimensional Fourier transform only really works if you have a flat metric on that space
What the hell does the metric of space-time have to do with this? When computing a fourier transform, we're not working in 3+1 dimensional space-time, we're working in either an N-dimensional (in the discrete case) or \infty-dimensional (in the continuous case) vector space; while that term contains the word "space" they DO NOT, in this context, have anything to do with Euclidean space or the Pseudo-Riemannian manifold that GR treats space-time as.
Which would be a bad thing, reading this kind of conversation is what makes this site worthwhile
Maybe calm down for a moment and try not being such a hot-headed ass. You seem to have missed the point entirely.
I’m well aware that these functions can be described as vectors in an infinite dimensional Hilbert space.
The problem I’m bringing up is that the domains of these functions (i.e. not the vector itself) typically have geometric properties we care about.
The problem is that if one has a manifold with a non-trivial intrinsic geometry, then functions defined on that manifold cannot be faithfully Fourier transformed without losing pretty much all geometrically relevant information.
It turns out that in some cases, there are generalizations of the Fourier transform of a function on a curved manifold, but in those cases, the domain of the transformed function is very different, typically having a higher dimensionality.
This is particularly relevant and problematic in physics, where the Fourier transforms of functions on spacetime are really important and useful, but dont work in curved spacetimes.
E.g. it’s a big problem when doing QFT on a curved spacetime that one cannot separate positive frequencies of a field from negative frequencies.
Still, as pertains to whether "the frequency domain is a real place", maybe sine waves are relevant as representing resonant frequencies of physical systems.
There also seems to be something fundamental about the way multiple radio frequencies can simultaneously propagate through a vacuum as long as they are different frequencies.
To me what you described sounds more like the Fourier series.
the delta with smallest frequency is the fundamental frequency, and the others are harmonics
when you do the inverse fourier transform on this train, each delta becomes a sinusoid
that's how you can write any periodic function as a sum of sinusoids, all of them multiples of the fundamental frequency
and that's the fourier series: it's just the fourier transform, followed by an inverse fourier transform, macroexpanded
but the fourier series only work for periodic functions, because only periodic functions have a bunch of isolated, periodic deltas as its fourier transform
so the fourier transform is only half the step of a fourier series (to write down the series you also need the inverse fourier transform) but, at the same time, the fourier transform is a generalization of the fourier series, because it works for nonperiodic functions too
But if those frequencies span a whole continuum (rather than frequencies f, 2f, 3f, 4f..), that is, an uncountable set of frequencies, then this signal is non-periodic and we can't talk about a Fourier series anymore, we must use the Fourier transform
Nitpick, but this isn't true. If my signal is a linear combination of two sinusoids - one at frequency 3 and the other at frequency 5, then there is no "fundamental" frequency when you do the FT.
"So energy is pumped into the system on the left, and is dissipated over here on the right"
"But the system is rotationally invariant, there is no left and right"
"I meant in frequency space"
"Oh I thought you were talking in real space"
"ARE YOU STUPID, WHO THE HELL THINKS IN REAL SPACE??!?"
Some academic communities are pretty dysfunctional.
is there even such a thing as left and right in frequency space?
It's an abstract representation, I'd it doesn't have any relationship to spatial dimensions in terms of left right up down
Convolution turns into multiplication, differentiation wrt time of the complex exponential turns into multiplication by j*omega. I don't know about you, but I'd rather do multiplication than convolution and time derivatives.
As a corollary, once you accept "we use the Fourier representation because it's convenient for a specific set of common scenarios", the use of any other mathematical transform shouldn't be too surprising (for other problems).
Technically it's a specialized case of the laplace basis, right? I was always surprised that lots of courses jump directly from the (bilateral) fourier transform to the unilateral laplace transform without proper analysis of the most general case that is the bilateral laplace transform: https://en.wikipedia.org/wiki/Two-sided_Laplace_transform
From an circuit analysis standpoint (your problem may be different), but exponentials that decay over time ("a" is negative) corresponds to loss in a circuit, whereas exponentials that grow over time ("a" positive) correspond to something blowing up (this is really a nonphysical result but generally means a circuit is going to oscillate on its own, without a source driving that response). I mostly do electromagnetics/passive RF types of problems, in which you generally want everything to be low-loss. In that case Fourier is perfect, especially since I typically care most about steady-state behavior.
There’s an optics experiment I did, bloody fiddly, where a picture goes through some lenses, and there’s a plane of the frequencies, and it goes through further lenses and is projected on a screen.
By blocking out areas in the frequency plane, you can change the image. It was extremely fiddly so huge thanks to Dr Bruce Sinclair at St Andrew’s.
Physics lab work is where you get to see how things work, although if you go through the theory several months after you’ve done the lab work you’re a bit lost.
Doesn't that always happen, which is why aperture limits resolution, why you get diffraction spikes with reflecting telescopes, and so on?
It is often used to determine the frequency of a period signal when you don't have fixed measurement interval, like in astrophysics.
https://iopscience.iop.org/article/10.3847/1538-4365/aab766 has a general introduction, and https://docs.astropy.org/en/stable/timeseries/lombscargle.ht... is a nice intro to using it in the astropy library (in Python)
But Prometheus data isn’t really a sampling interval. Even if each machine in you cluster is reporting on an interval, they aren’t synchronized.
So how do the first bone cells know to start becoming bone?
Edit: To expand on this, to interpret the cochlea as a fourier transform is to make the same mistake as thinking eyes have cone cells that respond only to red, green or blue light. The reality is that each cell has a varying reponse to a range of frequencies. Cone cells have a range that peaks in the low, medium or high frequency area and tails off at the sides. Cochlear hair cells have a more wavelet-like response curve with secondary peaks at harmonics of their peak response frequency.
Caveat: I'm not an expert in this, only an enthusiastic amateur, so I eagerly await someone well-akshuallying my well-akshually.
Could you expand on this a bit please? Especially as it relates to the Laplace transform.
Only a full Fourier transform, which has an infinity of outputs, could have (an infinite number of) filters with an infinitely narrow bandwidth, but which would also need an infinite time until producing their output.
So what you have said does not show that the eye cone cells do not perform a Fourier transform (more correctly a partial expansion in Fourier series of the light, which is periodic in time at the time scales comparable to its period).
The right explanation is that the sensitivity curves of the eye cone cells are a rather poor approximation of the optimal sensitivity curves of a set of filters for analyzing the spectral distribution of the incoming light (other animals except mammals have better sensitivity curves, but mammals have lost some of them and the ancestors of humans have re-developed 2 filters for red and green from a single inherited filter and there has not been enough time to do a job as good as in our distant ancestors).
At any rate, the point is that the frequency domain matters a lot, since our brain essentially receives sound data converted to the frequency domain in the first place...
By staring at the matrix, I guessed a pattern and algorithm already known according to https://en.wikipedia.org/wiki/Walsh_matrix:
> The sequency ordering of the rows of the Walsh matrix can be derived from the ordering of the Hadamard matrix by first applying the bit-reversal permutation and then the Gray-code permutation:
I would argue that despite this fact the frequency domain and by extension the FT is special compared to many other transforms, because we can actually observe them in nature. Two examples: a lens will perform a 2D FT of an input image on a collimated beam, we can observe this with e.g. a screen. Second example, we can measure the wavelength (or frequency) of light by projecting the output of a grating or prism onto a ccd again a direct measurement of the FD (similar measurement can be done for RF waves).
I can think of lots of other places I'd use fourier analysis (at least qualitatively as with doing diffusion modeling in my head) but you're right that sinusoids are more physically "real" whereas being possible to represent in any basis set is more "valid" if that makes any sense.
Not quite sure what the right word is on this one, but I agree "real" kind of suggests real oscillators underlying the phenomena. Square waves are less physical because of discontinuities in both the signal and derivative; nature really doesn't care for discontinuities.
The physics result is actually probably a consequence of that.
At the end of the day the whole lesson of modern math is that its useful to view things from many perspectives.
The Gibbs phenomenon, for example, falls out naturally from the IFT of a frequency response where all the frequencies above some cutoff are zero.
I'm curious how the square wave frequency domain would describe the Gibbs phenomenon -- I think you'd have harmonics of the fundamental square frequency showing up as if the system were nonlinear.
If you have a signal with a single narrow peak in the time domain you can represent it in a very compact or sparse way just using a delta at the position of a peak. If you try to represent it in the frequency domain you will not find such compact representation. Similarly if you have a sinusoidal signal in the time domain, you won't get a compact representation there, but you will get it in the frequency domain where you just have a couple of deltas.
Time and frequency are two ways of representing the same thing. Sometimes it's easier to represent something in one domain, sometimes it's easier in the other domain.
It can be proven that anything bounded in the time domain will be unbounded in the frequency domain and viceversa. So compact stuff in one domain always spreads when represented in the other domain.
Beautifully, quantum mechanics tell us that position and momentum are conjugated variables (like time and frequency in the example above) and therefore if something has a bounded position (we know where it is) its momentum will be unbounded (we won't know its speed), and viceversa.
That's the main idea of Heisenberg's uncertainty principle.
Now I consider them to be just as "real" as the integers - which is not at all. Both just human-invented concepts with no fundamental physical basis.
As you point out - useful, though!
Both are abstractions. That means that they are properties of real physical objects, which are obtained by ignoring all the other properties of those physical objects that are irrelevant in the context of the application.
Therefore an abstract property is an equivalence class of physical objects, where all their other properties are ignored, so they are equivalent if they have the same value for the property of interest.
Non-negative integer numbers are equivalence classes of collections of physical objects, integer numbers are equivalence classes of pairs of such collections.
The imaginary unit is the equivalence class of all rotations by a right angle in the 2-dimensional space. Humans, like many animals, have an innate ability to recognize right angles, like also certain small numbers, so looking around you can perceive as easily all imaginary units like all numbers 3.
The complex numbers are the equivalence classes of all geometric transformations of the 2-dimensional space that can be decomposed in rotations and similarities (a subset of the affine transformations). In contrast, the 2-dimensional vectors are the equivalence classes of all translations of the 2-dimensional space (another subset of the affine transformations).
All the things that are equivalent from the point of view of an integer number or a complex number, so they are the basis from which such numbers are abstracted, are things that you can see with your own eyes in the physical world (similarity transformations appear in optical projections, e.g. in the shadows of physical objects, and the eyes are based on them).
You can do QM without complex numbers as people are used to use them.
But it gets really awkward really fast.
Real-valued (operator) fields (like the photon field) do not have distinct particles (a photon is its own antiparticle)
I can elaborate more if needed.
There're not only no empirical but no theoretical grounds to believe anything like this. The mind is almost certainly wholly defined by the physical processes of the brain, in the same spacetime realm all other known physical processes reside in.
I believe this is basically just the dualism vs. materialism debate on consciousness. Consciousness is a fascinating topic. There‘s plenty of paradoxes or thought experiments to fry your brain on. It‘s not just about the electrochemical processes in the brain. It‘s about identity, the continuity thereof, etc.
Is there a good collection of these in one place?
Just because something is unmeasurable doesn’t mean it cannot be proven wrong.
Just because something is derived from philosophy doesn’t mean it cannot be proven to be wrong in the real world.
> Science can only help us understand things within spacetime. Anything beyond that is philosophy.
Even if one takes this statement to be correct, it doesn’t imply that any specific philosophical idea about “beyond space time” is correct.
And frankly even the “philosophy” of an after life can easily be dispensed with. There’s absolutely no reason to suggest an after life, or a duality between the body and “identity” exists other than “we would like to believe so”.
It’s not just awful science but also bad philosophy.
Also why should you restrain from pointing out weaknesses in other people's comments due to the fear of negative karma? Karma is meant to be burned.
Do they and the people upvoting them not realize that it’s a blog post?
I do know I've thought about this, and even had one of those "moments of realization" on the drive home from my first lecture on DCTs, where I thought I could transform all of human history from time domain to frequency domain and how this would bring out certain patterns and truths that could not be understood otherwise. I swear I wasn't on drugs! (Though the lecture was in fact given by Prof. Marshall of Rutgers, the father of the lyricist for the rock band Phish, at whose concerts I have imbibed certain substances... but I digress.) The frequency domain is just as "real" as any other mathematical construction that can help us understand everything.
We can have a standing wave, e.g. vibrating string. Frequencies then translate to concrete place in space, such as the nodes where the string appears to be still, (like the exact middle if it is excited with the second harmonic).
This is conceptual and theoretical, but potentially has immediate application for computing a better offset of a cubic Bézier, used for stroke expansion.
If this sounds intriguing, a good starting point is the Zulip thread[1] I'm using to write down the ideas. I'd especially be interested in a collaborator who has the experience and motivation to coauthor a paper; I can supply the intuition and experimental approach, but the details of the math take me a long time to work out. (That said, I'm starting to wonder if engaging that slog myself might not actually be a good way to level up my math skills)
[1]: https://xi.zulipchat.com/#narrow/stream/260979-kurbo/topic/E...
Simply look around and tune it. /
Anything you want to - tune it. /
Want to change the channel? nothing to it.
There is no SDR /
To compare with foreign number stations /
Casting there, you'll be free /
If you tune to ninety eight point three
>"This is known as the Walsh matrix. It essentially consists of square waves running at different speeds, although with some complex symmetries thrown in. And yes: every multiplier is just a +1 or a -1, so the computation boils down to flipping some signs in the input data and then summing the results.
The matrix looks fairly trivial, but its design is involved. To capture all frequency and phase information, the rows have increasing sequency — that is, each next row has just one more sign flip than the one before. Further, the pattern is carefully engineered to ensure orthogonality — the fragile input-output symmetry that allows seamless conversions back and forth between the frequency representation and the original time-series data.
[...]
>"Hadamard [Matrices]:
At a glance, the plot looks chaotic, but it’s simply a reordering of the Walsh layout. For example, rows #1 and #8 are swapped; the same goes for #1 and #15. And unlike Walsh, this fractal-esque pattern is actually fairly easy to create from scratch."
I did not know any of this before!
Anyway, an excellent article! (Applications of Walsh & Hadamard Matrices in Physics and in modern technology -- are too numerous to count!)
Related:
“I'm using C in lieu of imperative pseudocode. The benefit it has over pure pseudocode is that it compiles and doesn't rely on some syntax I pulled out of nowhere. The benefit over higher-level languages is that it shows the mechanics of the calculation in a way that's trivial to transpose to any other dialect. This can't be said about code that uses cute but language-specific array manipulation and iteration idioms, e.g.:
WalshMatrix = Nest[ArrayFlatten@{{#, #}, {#, -#}} &, 1, #] &;”
Fourier transform is not the same thing as the DFT to begin with. Thinking of the Fourier transform as just the decomposition of signals into sine waves may not be the right way of looking at it either.
A very useful feature of Fourier transform (and Laplace transform, more generally) is to convert ordinary (and some partial) differential equations to polynomials, which turns solution of differential equations to polynomial algebra. This was also the original goal of Mr. Fourier for solving the heat equation.
Are there other transformations that can do that?
But many places are real without being special - other than to those who make special use of them!
I'd argue a real place is one that affords the operators that allow us to inhabit and interact there -- stuff like object permanency, adjacency and distance. If things can be organized and sustained there, are they not real?
It's fun to imagine what kinds of structures can inhabit the frequency domain - or any other.
For example, one thinks of tug C major triad as C E G, rather than mixed sound that comes out of C E G together.
This question has a long and fascinating history.
So, I would suggest a better question is: where can I find an introduction to this stuff that is better aligned with my current understanding and knowledge. To that end, here's a great intro book: https://www.dspguide.com/
I read that cover to cover when I was 17 and I would not have been described as particularly mathematically gifted.
The things you can hear scale pretty well all the way down to DC, and all the way up to daylight (or beyond, I suppose).
After spending a few (or dozens, or hundreds) of hours tinkering with audio-range frequencies for fun, then maybe you'll have a real place with which to associate the maths that are involved in electronics and programming, and having that place might make that math (and whatever is needed to accomplish it) feel lot more worthwhile to actually-learn.
I agree. Start with small hobby projects using what you already know.
They should be things where you get immediate feedback, like graphics or sound. That makes them very fun to work and iterate on. It's addicting when you see/hear the output, make a change in a few seconds, and immediately see/hear the results of your change. That's the stuff that keeps you up all night at the computer.
You'll start hitting barriers where you need more math to get better results. Then you'll be much more motivated to start learning, and it will be easier to retain the knowledge when you actually use it in practice.
For instance, use Python to output audio samples. Start with simple sine wave tones, colored noise, and work your way up to implementing simple FIR and IIR filters to modify input audio like voice and music. Use Audacity to see the change between input and output as a spectrogram.
If you're into music, use an existing Python library to read MIDI files with songs you like. Generate audio output files for those songs. First with just sine waves for the notes, then you can start emulating digital and analog synths. Write code that takes sampled audio files and emulate different guitar sound effect pedals or tube amplifiers.
Don't look too much at the Greek letters, look at the tables and code instead. A DCT is just a bunch of multiplications, additions and a lookup table. The approach to derive the Walsh-Hadamard transform is actually very computer sciencey. It is made using a recursive algorithm, it then takes advantage of the fact that a+b==b+a to reorder the rows. There is even a trick that uses a bitwise AND.
As a programmer I find it much easier to start from here: tables, loops, and simple operations like multiplication and addition. And when I finally understand how a computer does the thing, maybe go back to the maths to see if I can get more insight.
In the end, the only appropriate answer is "get good at maths". The question is how to achieve that. And if you have a programmer mindset like I do (and I guess like many people here) and struggle with maths, I recommend trying the bottom up approach. Write the code, have it actually show visuals and play sounds, play with the parameters and see what changes and what doesn't change.
The other part is overcoming the language barrier, I am still struggling with that. Mathematicians use barbaric name like eigenvectors for "stuff that won't change", they call "integration" an addition in a loop and write it with a weird sigma thing, or a weird "S" thing when there are a lot of iterations over very small numbers. But in the end, it is not that different from reading Perl :)
uh,
I’m not sure what statement that others might believe, the article is contradicting?
The Fourier transform has some nice properties that other bases don’t.
One way to view the Fourier transform, is as decomposing “the space of functions on [some domain], regarded as a representation of [the group of translations in that domain]”, into one-dimensional sub-representations. (Or, if you want to stay over the real numbers instead of complex numbers, then two dimensional representations)
I don’t know that there’s a nice analogy of this for the basis described in the article.
Maybe there is? Like, maybe if one uses some group with 2^n elements other that Z/((2^n)Z) , maybe (Z/(2Z))^n, then maybe the basis you would get would be like the one shown?
Idk.
But I don’t see this as showing frequency space as less real.
it would
The title prompted a science (or, maybe, philosophy of science) question, but the article give only applications.
Maths don't have to be real to be useful.
Seahorse Valley is more place like if you want to debate the reality of these things https://www.mrob.com/pub/muency/seahorsevalley.html
TLDR: Fourier transform is an approximation. We can come up with other approximations, for example by square waves.