I want to start with a bit of caution: In this post you can find the definition of the 4D Fourier series. It looks a lot like the definition as on the complex plane. But I still did not prove all those convergence questions. And I also do not remember very much from the time when I had that stuff as a student (that was about 30 years ago). So I don’t know if I will be able to make such proofs about convergence and what kind of functions you can use to make a Fourier series from etc etc.
Yet in this post I define a set of possible signals that I name ‘pure tones’ and these clearly have a 4D Fourier expansion because by definition they have a finite number of non-zero Fourier coefficients. Of course when you only have a finite amount of non-zero coefficients, you don’t have any kind of convergence problem. So for the time the convergence problems are avoided.
In this post, number 154 already, I hope I demenstrated enough that the basis functions used in the definition are all perpendicular to each other. After all that was a nasty hurdle we met when it was tried with just the four coordinate functions of the 4D exponential curve as our basis vectors. So we do not meet that problem again using the exponential curve as a whole. If I denote the exponential curve as f(t), the basis functions we use are basically f(nt) where n is a whole number. Just like in the previous posts I always use the notation f(t) when the period is related to the dimension and g(t) when the period is different. Here I use of course a period of two pi because that is convenient and it makes the coordinate functions more easy to write out: the first one is now cos(t)cos(2t) and the other three are just time lags of the first one. But if you want to write g(t) as an exponential, because of the period it now looks a bit more difficult compared to just e to the power tau times t.
For myself speaking I have no idea at all if crafting a Fourier series like this has any benefits of using just the definition as on the complex plane. After all I only have more or less basic knowledge about the use of Fourier series, so I just don’t know if it is ‘better’ in some regards and ‘worse’ in others.
At last without doubt under my readers there will be a significant fraction that wonder if those 4D complex number system is not some silly form of just the complex plane? After all if that 4D space is based on some imaginary unit l with the property that now the fourth power l^4 = -1, how can that be different from the complex plane? The answer to that lies in the logarithm of the first imaginary unit l. If this 4D space was just some silly extension of the complex plane, this log of l should be nicely bound to i pi/4 where i is of course the imaginary unit from the complex plane. But log(l) is the famous number tau because with that you can make the exponential curve f(t) = e^(tau t). Basically the main insight is that i pi/4 makes the complex exponential go round with a period of four because i^4 = 1 and the 4D number tau makes the exponential curve go round with a period of 8 because l^8 = 1.
This post is six pictures long, all 550×775 pixels in size.
As usual I skipped a lot of stuff. For example, how did Fourier do it? After all at the time all this stuff with inner products was poorly developed or understood. That alone would be a cute post to write about. Yet the line of reasoning offered by Joseph Fourier was truly brilliant. In case you are lazy or you want to avoid Google tracking you, here is a link to that cute symbolab stuff: symbolab.com Link used: https://www.symbolab.com/solver/fourier-series-calculator
Ok, that is what I had to say for this tiny math update.
This evening I brewed a fresh batch of wort, it is now cooling and tomorrow when it is at room temperature it can go into the fermenter bottles. Everything is relaxed around here, in the last week in my city only four cases of COVID-19 into the hospital.
Orginally I planned to define the 4D way of taking a Fourier series in this post, but while writing this post I realized to would become too long. So this post is seven pictures long (all 550×775 pixels) therefore I hope it does not cross the length of your attention span. For myself speaking always when I read some stuff that is ‘too long’ I start scrolling till the end and as such you often miss a lot of important details.
An important feature of exponential curves is that you have those de Moivre identities that come along with all those exponential curves. In this post I did not prove the 4D version of a de Moivre identity for the 4D complex numbers, but I give some numerical evdence. It has to be remarked that when I wrote the 20 posts around the basics of the 4D complex number system, I did not include the de Moivre identity. So that is more or less an ommision. On the other hand it is of course much more important to be able to find the exponential curve in the 4D space that is the basic material needed to write down such a de Moivre identity.
I categorize these posts about the 4D Fourier series also under ‘integration’. Not that I have many fantastic insights about integration but the reason for this category is much more down to earth: You have to perform an integration for every Fourier coefficient you calculate. In case you missed it: this year I finally wrote that post about how to define integration in the 3D complex numbers. Use the search function of this website in order to find that post in case you are interested.
Ok, that is what I had to say. Let us go to the beef of this post number 153 and that is of course the seven pictures that are all hungry for your attention. Here are the seven pictures:
There is an online Fourier series applet as it is named in these years. In the past we did not have applets but only computer programs. Anyway it is important those applets are there so I want to give a big thumbs up to the people that maintain that website! Here is a link to the Fourier thing:
At last I want to point at the importance of such free websites. It is very good if a society has enough of this kind of ‘free stuff’. For example when in the year 2012 I picked up my study of the 3D complex numbers for me it was very important that there were free online applets for the logarithm of a matrix. Without such a free website I would have taken me many more years to find those exponential circles & curves. Or I would not have found them at all because after all my biggest breakthrough was when I did read that numerical evidence from log j on the 3D complex numbers: why are the two imaginary components equal I was just wondering? Later I found how crucial that was: only if the imaginary components are equal the eigenvalues are purely imaginary in the complex plane.
So I am breaking a lance for free websites where you can find good applets (read: computer programs) that help your understanding of math.
Ok, that’s it for this post. In the next post I will finally give that definition and after that I do not have a clue. I still do not have any good proof for convergence of these 4D Fourier series so we’ll see. Till updates.
Warning: This post contains stuff that is not correct! Yet I decided to post it anyway so you can see that crafting math is also just keep on working until you have it right. The reason it does not work in the post below is that the basis functions I use are not all perpendicular to each other. And if you want to reconstruct a function or a signal s(t) with the basis functions as mentioned below, you will get weird overlap and the end result will not be correct.
Ok, that was a bit frustrating. But all in all I don’t have much reputation damage because more or less instantly I found another way of crafting the 4D Fourier series and that seems to work perfectly. So compared to the professional math professors who at one point in time accepted the quaternions and together with that stupid theorem of Frobenius concluded that 3D or 4D complex numbers are not possible, at least I don’t look that retarded. Sometimes I can be stupid too but at least it does not last for over one century. And may be that is also the reason that professional math professors absolutely do not want to talk about my work on 3D & higher dimensional number systems; admitting that you have been stupid for over one century is of course not an easy thing to do. And given the fact I am now unemployed for 17 or 18 years, rather likely the professional professors would rather be eating dog shit compared to speaking out my name… Once more we observe that in this world there is never a shortage of idiots.
In this post I use the coordinate functions of the exponential curve in the 4D complex numbers but I changed the period to 2 pi instead of a period of 8 that comes along with taking the log of the first imaginary unit. I also would like to mention that I use the so called modified Dirichlet kernel and because that kernel originates from Fourier analysis you must not get confused by the name ‘Dirichlet kernel’. The modified kernel is important (anyway for me) because it spits out all those coordinate functions for making exponential circles and curves in all dimensions possible. While if memory serves, the use of the Dirichlet kernel inside Fourier analysis is for using it in proofs of convergence. But may be I remember that completely wrong, after all it was about 30 years ago that I studied the Fourier stuff for the last time. The last two weeks were pleasant from the mathematical point of view, all that old Fourier stuff that somewhere still lingers around in my brain. But so much is gone, what is that Gibbs overshoot? Is that when a male math professor has his yearly orgasm? And what was the Parcival identiy? I don’t have a clue whatsoever. This post is 7 pictures long, four are of 550×775 pixels and I had to enlarge the other three to a size of 550×850 pixels. So it is not a mess like the previous post where I just enlarged the pictures on the fly until all that text was there. Here it is:
Likely all those basis functions have this problem, if you take the inner product of an arbitrary basis function against the same basis function with three times the speed, it is not zero. And as such it is not perpendicular…
For people who have never heard of inner product spaces done with functions I found a cute pdf where a lot of the basics are explained.
I would like to be the 4D Fourier stuff done in a correct manner in the next post but sifting through what I wrote on 4D complex numbers I realized I never wrote about a de Moivre identiy for the 4D exponential curve. May be I will publish that in a separate post, may be not. Anyway, have a good time and see you in the next update.
A lot of math professionals rather likely still think that 3D complex numbers do not exist, may be for reasons like there are non-invertible numbers or whatever what other reason they have. This post more or less proved such views are nonsense; for example a lot of math on the 2D complex plane does not rely on the fact it is a field (and as such only division by zero is forbidden).
But on the 3D complex and circular number spaces indeed it brings some complications if you have non-invertible numbers in the function you want to integrate over a particular curve. And I have to say that problem could be solved by using the special properties that those numbers have. In this post I only show some examples with the non-invertible number alpha (alpha is the midpoint of the 3D exponential circles and all multiples of alpha are also non-invertible so the line through 0 and alpha are all not invertible).
For me writing this was a good distraction away from all that negative news we have day in day, all those countries reporting daily death toll can make you a bit depressed… So when I am through with the daily news I always do some other stuff like calculating a few of such integrals. That is a very good antidote against all that bad news. After all there is not much gained if you constantly think about things you cannot change at all.
This post is relatively long; at first I crafted 12 pictures but it soon turned out that was not enough. So while filling the 12 pictures with the math and the text I expanded some of the pictures so they could contain more math & text. That was not enough and in the end I had to craft two more background pictures. All in all it is 14 pictures long, that is a record length for this website.
If in your own mathematical life you have performed contour integration in the complex plane, you must be able to understand how this works in the 3D spaces. And for those who have done the thing known as u-substitution on the real line: it is just like that but now this u thing is the parametrization of a path. All that stuff below with gamma in it is either the path or the parametrization of that path. Please remark that you must use the complex or the circular multiplication on 3D, just like integrating over a contour in the complex plane uses the 2D complex multiplication.
In case if you are not familiar with the number alpha that is found at the center of the exponential circle, use the search function of this website and for example look up ‘seven properties of the number alpha’.
I hope I have removed all faults, typo’s etc so that later I do not have to repair the math because that is always cumbersome. Here we go: 14 pictures long so this is hard to grasp in detail in just a few hours. But it is beautiful math & that is why I do this. For me math is a lovely hobby.
Enough of the blah blah blah, here we go:
Ok, let´s first hit the button ´Publish´ and see what will happen… It looks all right but a day after first publication I realized there was some missing text. It turned out I had to rename picture number 2 and now every thing was like it was planned.
Later I will flea through the rest of the text, if needed I will post more addenda. For the time being that was it so till addendums or till the next post.