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I've always grappled with anything related to Fourier since my undergrad days. Recently, when revisiting why I learned what I did, I discovered how Fourier's desire to understand the flow of heat through a solid body led to the creation/discovery of Fourier Series and correspondingly the Fourier Transform.

However, I've never been able to make the mental leap from the heat equation to the creation of Fourier Series.

Are there any good sources (references, books, videos (most preferable) etc.) that provide a walkthrough to help "discover/create" the findings of Fourier by yourself?

This is purely an intellectual exercise out of curiosity - it may not be worth it. Still, I'm curious to be able to understand it "naturally" vs being told it is so.

My expected "thought experiment setup" is a thin rod with thermometers at regular intervals and using that to derive what Fourier did - not sure if this is even possible, but the idea is to go from the simplest abstraction to the actual concept at hand. It's okay if things are mixed with modern calculus to help understand it better since at the time it was still in flux.

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§1.1 (+ supplement) of Bressoud’s A radical approach to real analysis, recommended here just recently, does pretty much exactly what you want.

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  • $\begingroup$ For digging more deeply, the OP can see the references I gave in my comment to How did Fourier arrive at the following regarding his series and coefficients? Also, the book Who Is Fourier? A Mathematical Adventure is possibly worth looking at. I don't have a copy of this book, but two or three times since it appeared I've looked through copies I've seen in libraries. But as far as occurs to me right now, I think Bressoud's book is the best place for the OP to start. $\endgroup$ – Dave L Renfro Nov 14 '18 at 11:55
  • $\begingroup$ FWIW my copy of the book arrived today. Looks like it may have what I’m looking for and a lot more that I’m intrigued about. Thank you!! $\endgroup$ – PhD Nov 16 '18 at 6:54
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Contrary to what the name suggests, Fourier series were not invented/discovered by Fourier. They were considered by Euler and Bernoullis, in relation to the one dimensional wave equation, not the heat equation. This early story is described for example in the papers by Luzin in Amer. Math. Monthly:

Luzin, N. Function. I. Amer. Math. Monthly 105 (1998), no. 1, 59–67.

Luzin, N. Function. II. Amer. Math. Monthly 105 (1998), no. 3, 263–270.

(Hovewer, Fourier integral and theta-functions are Fourier's inventions). The book of Fourier is translated into English, and it is still a very interesting reading. In it Fourier gave a systematic theory of solving PDE's by the method of separation of the variables, and after its publication, Fourier series became a general tool in mathematics and physics. So the names Fourier series and Fourier analysis are well justified.

Remark on comments. Fourier did not establish with a rigorous proof any general criteria of representation of functions by Fourier series. His book was criticized as non-rigorous, and this substantially delayed its publication. However he made very convincing arguments. He also checked some of his results by actual experiment with heating various bodies.

Fourier was a scientist, not only a pure mathematician.He said that the main goal of mathematics is a profound study of nature. One of the principal motivating questions for his study of heat was determination of the age of the Earth. This was later done by Thomson (Lord Kelvin) using Fourier theory.

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    $\begingroup$ That is very true, and e.g. Lebesgue (1906, pp. 19, 22–23) called the integral formulas for the Fourier coefficients “formulas of Euler and Fourier”. However, I don’t think anyone until Fourier believed that an “arbitrary” function has a trigonometric expansion, as needed to match “arbitrary” boundary conditions in PDEs. $\endgroup$ – Francois Ziegler Nov 14 '18 at 14:58
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    $\begingroup$ (Euler’s Fourier series can be seen in (1749, p. 35; 1753, p. 81; 1798, p. 116).) $\endgroup$ – Francois Ziegler Nov 14 '18 at 15:19
  • $\begingroup$ @Francois Ziegler: some believed, and there was a long and heated discussion about this. See Luzin's papers mentioned in my ans. Neither did Fourier prove this. But he made a strong case:-) $\endgroup$ – Alexandre Eremenko Nov 14 '18 at 18:39
  • $\begingroup$ @AlexandreEremenko - you're right. I am aware of it's "prior existence" w.r.t. to the heat equation and Euler's work. Was trying to keep my question less verbose and get to the meat of what I wanted to get to. However, the references you've suggested seem very promising. So I'm glad I wasn't entirely accurate :) $\endgroup$ – PhD Nov 14 '18 at 18:58
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    $\begingroup$ @PhD : Actually, the first proof of convergence the Fourier series-- which is typically attributed to Dirichlet--was found in Fourier's original manuscript. And it is correct. Take a look at my answer here: hsm.stackexchange.com/questions/7622/… $\endgroup$ – DisintegratingByParts Dec 6 '18 at 5:50
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The book Introduction to the Theory of Fourier's Series and Integrals by H. S. Carslaw answers your questions in the first chapter on the History of this subject. Many commonly held false beliefs are debunked in his first chapter, including the idea that Fourier failed to give a rigorous proof of convergence. Another common false belief is that Fourier discovered the Fourier coefficients. The "orthogonality" conditions were discovered by Clairaut and Euler. This is a fascinating bit of History that is often incorrectly cited.

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    $\begingroup$ This is a nice and remarkably detailed historical survey that I suspect is not as well known as it should be. I've had this book (Dover edition) since the late 1970s, so I've been well aware of it. I found it very helpful in my initial explorations of the history of nowhere differentiable functions (Spring 1992), before I dug more deeply into many of the papers by Philip E. B. Jourdain and others (and original sources) (continued) $\endgroup$ – Dave L Renfro Dec 6 '18 at 7:55
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    $\begingroup$ during the next few months, for historical background material that I included in my 1993 dissertation. A survey similar to Carslaw's (also a historical introductory chapter to a textbook that has been reprinted by Dover, also written by someone with the historical and mathematical breadth to do so successfully, but the focus is more on integration theories) is the Historical Introduction (pp. vii-xxvi) in An Introduction to Analysis and Integration Theory by Esther R. Phillips (I got the 1984 Dover reprint sometime in the mid 1980s). $\endgroup$ – Dave L Renfro Dec 6 '18 at 8:07
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    $\begingroup$ I just noticed that the Historical Introduction to Phillips's book did NOT appear in the original 1971 edition, but rather it was newly added for the 1984 Dover Publications edition. $\endgroup$ – Dave L Renfro Dec 6 '18 at 9:23
  • $\begingroup$ @DaveLRenfro : I always appreciate references like yours. $\endgroup$ – DisintegratingByParts Dec 7 '18 at 16:58
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I know a book that is completely devoted to the history and development of trigonometric series (including of course Fourier series). But it is not in English, it is in Russian. It gives a rather detailed treatment on the method of various mathematicians who involved in this field. The title is "A.B Paplaukas Trigonometric series from Euler to Lebesgue" (Паплаускас А.Б. Тригонометрические ряды от Эйлера до Лебега). It has a free electronic version, so if you can read Russian, you can download it.

I like Russian math books. They are quite rigorous while stay away from heavy formalism. The book I recommend do not spend times on chatting about irrelevant stories. It is very direct in style.

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The OP wrote:

Are there any good sources (references, books, videos (most preferable) etc.) that provide a walkthrough to help "discover/create" the findings of Fourier by yourself?

I think chapters 8 and 9 of 17 Equations That Changed the World by Ian Stewart provide such a walkthrough.

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For me, the "a-ha" moment with Fourier transforms was when I related it to resolving a vector into components. If you have a vector in some arbitrary direction and for convenience of calculation or representation you want to express it as the sum of scalar multiples of another pair of vectors (let's call them the "basis pair"), you use the dot product of that vector with each of the basis vectors (divided by the basis vector's magnitude) to find something that could be called "the shadow of that vector projected along the line of the basis vector".

In a simple case, the basis pair could be the horizontal unit vector and the vertical unit vector--that's what you're most familiar with--but any two vectors that don't lie along the same line can be used, you just have to project the vector you're looking at along the line of each vector in turn and see what a and b you need to use as a multiple to get a times {one vector} plus b times {the other vector} to add up to your original vector. Fourier transforms use a similar idea--"I've got this complicated wave form, let me see if I can break it up into a sum of multiples of simpler wave forms."

Think about the Fourier transform on a sound wave as an example. What you're basically doing is projecting the sound wave onto a pure 440Hz signal and saying "it's about this much middle A" and then projecting the wave onto a 466Hz signal says "and now about this much B flat". Continuing in this way you would have the tones on a piano as your "basis vectors" and you could reproduce the sound (approximately) by playing each key harder or softer according to what the calculation gave you at each frequency.

To me, this takes some of the "well how would anyone ever come up with this?" mystery out of it. The idea of representing something as a times this basis vector plus b times that basis vector is just being generalized to "I can represent this sound wave as a times this pure sine wave plus b times this pure sine wave, plus c times this pure sine wave" and so on.

I don't know if this is historically where the realization came from, but I think it answers your question in the sense of "how would I ever come up with this idea myself?". You come up with it by thinking of a curve as being the sum of components of an infinite-dimensional vector space. Don't let the "infinite dimensions" mess with your head--in practice, you just get it as close as you need it to be, like with digital music you take the natural sound and sample it over enough different frequencies (basis vectors) that it sounds ok when you play it back as x times this frequency plus y times that frequency plus z times that frequency, etc.

(This is how a Fourier transform on an input wave could be used to digitize music--you go from a natural wave to "play frequency x [a number, stored digitally] at volume y [a number, stored digitally] and, simultaneously, frequency z [a number, stored digitally] at volume v [a number, stored digitally]". You're just "resolving" the music into numerical volume multipliers of numerical frequencies.)

You don't get an exact fit for every possible curve but by using more and more "basis vectors" you get closer and closer to the exact curve.

The "piano keys as basis vectors" idea is not new to me--one of my physics professors showed us how you could yell into an open piano and the sympathetic vibration of the strings would make an approximate Fourier transform of your voice. See also this experiment along those lines in a YouTube video (narration in German but the piano "talks" in English): https://www.youtube.com/watch?v=muCPjK4nGY4

Other concepts in math have a similar basis (pardon the pun). Taylor series expansion and Bernoulli polynomials are similar things where you get close to some function or curve by approximating it with a bunch of scalar multipliers to something that approximates an infinite-dimensional vector space. The process of figuring out what the scalar multipliers are for a given curve usually looks something like the Fourier transform.

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