In "practice" it is likely "useless", because the functions that it can integrate that Riemann can't are just too funky to appear in practice :-)
Its value is much more indirect and subtle, as in "it serves as a solid basis of quantum mechanics" due to the definition of Hilbert spaces.
is:
- complete under the Lebesgue integral, this result is may be called the Riesz-Fischer theorem
- not complete under the Riemann integral: math.stackexchange.com/questions/397369/space-of-riemann-integrable-functions-not-complete
And then this is why quantum mechanics basically lives in : not being complete makes no sense physically, it would mean that you can get closer and closer to states that don't exist!
A measurable function defined on a closed interval is square integrable (and therefore in ) if and only if Fourier series converges in norm the function:
norm sequence convergence does not imply pointwise convergence by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Integrable functions to the power , usually and in this text assumed under the Lebesgue integral because: Lebesgue integral of is complete but Riemann isn't
is by far the most important of because it is quantum mechanics states live, because the total probability of being in any state has to be 1!
has some crucially important properties that other don't (TODO confirm and make those more precise):
- it is the only that is Hilbert space because it is the only one where an inner product compatible with the metric can be defined:
- Fourier basis is complete for , which is great for solving differential equation
Some sources say that this is just the part that says that the norm of a function is the same as the norm of its Fourier transform.
The comment at math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the bijection statement is an easy consequence from the norm one, thus the confusion.
TODO does it require it to be in as well? Wikipedia en.wikipedia.org/w/index.php?title=Plancherel_theorem&oldid=987110841 says yes, but courses.maths.ox.ac.uk/node/view_material/53981 does not mention it.
As mentioned at Section "Plancherel theorem", some people call this part of Plancherel theorem, while others say it is just a corollary.
This is an important fact in quantum mechanics, since it is because of this that it makes sense to talk about position and momentum space as two dual representations of the wave function that contain the exact same amount of information.
Ciro Santilli has become slightly obsessed with this story, and the main mastermind Ross Ulbricht.
The best article available so far is: www.theregister.co.uk/2019/01/29/how_i_caught_silk_road_mastermind (archive) which summarizes what one of the investigators said in a 2019 French computer security conference.
The key living posts are:
- stackoverflow.com/questions/15445285/how-can-i-connect-to-a-tor-hidden-service-using-curl-in-php (archive) which was originally asked under the real name, and then the username was changed to "Frosty", which matches one of the server's logins after the laptop was captured
- altoid early Silk Road mention: bitcointalk.org/?topic=175.70;wap2 (archive)
The big question is of course how libertarian free market ideologically motivated the website was, and how purely criminal greed it was.
The magnitude of the early operational security mistakes does make Ciro think that Ross did it "because he could" and "for the lolz" in a real world Breaking Bad way.
The entry in Ross' diary does resonate a lot with Ciro and any entrepreneur, full diary at: www.wired.com/2015/01/heres-secret-silk-road-journal-laptop-ross-ulbricht/ (archive).
[i]n 2011," [I believe I will be] "creating a year of prosperity and power beyond what I have ever experienced before,Silk Road is going to become a phenomenon and at least one person will tell me about it, unknowing that I was its creator."
Having this kind of feeling, is the greatest thing any human can have, and what motivates all great things.
Capitalizing in illegal things though is a cheat, big things take longer than a few years to reach, but reaching them is that much more satisfying as well.
Other interesting quotes:which Ciro also feels, see don't be a pussy, and:
Everyone knows I am working on a bitcoin exchange. I always thought honesty was the best policy and now I didn't know what to do. I should have just told everyone I am a freelance programmer or something, but I had to tell half truths. It felt wrong to lie completely so I tried to tell the truth without revealing the bad part, but now I am in a jam. Everyone knows too much. Dammit.
Also very worth reading is the San Francisco flat mate account: www.vice.com/en_us/article/ae3q8g/my-roommate-the-darknet-drug-lord (archive).
The murder for hire allegations are also interesting: mashable.com/2013/10/03/silk-road-hits, he paid 80k dollars to undercover DEA agents!
Except for the fact that Ross was an 80 million Dollar drug lord, those accounts sound exactly like what you would expect from any other nerdy startup founder! The:
- "just do it" strategy effectively going to a minimal viable product (manual transaction management!), while making many mistakes along the way, including hiring mistakes and successes when scaling is needed
- the hardship of self bootstrapping your own social network (here with some kilos of mushrooms)
- the variety of periods, from relatively calm, to hair pulling stress during big changes
It is also amusing to see very concretely the obvious fact that the FBI can get a subpoena for all accounts you ever had, e.g. they knew his laptop model from Amazon and brought a corresponding power cable to the arrest! If you are going to be a cyber criminal, don't use your real name, ever!
Should justice be blind? Maybe. But it does hurt for mere non-blind men to see it sometimes. Especially when drug liberalization is involved.
Main motivation: Lebesgue integral.
The Bright Side Of Mathematics 2019 playlist: www.youtube.com/watch?v=xZ69KEg7ccU&list=PLBh2i93oe2qvMVqAzsX1Kuv6-4fjazZ8j
Solving partial differential equations with the Fourier series by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16See: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from heat equation solution with Fourier series.
Separation of variables of certain equations like the heat equation and wave equation are solved immediately by calculating the Fourier series of initial conditions!
Continuous version of the Fourier series.
Can be used to represent functions that are not periodic: math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.
Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.
As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.
Lecture notes:
- www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf Lecture 2: 2D Fourier transforms and applications by A. Zisserman (2014)
How the 2D FFT works by Mike X Cohen (2017)
Source. Animations showing how the 2D Fourier transform looks like for simple inpuf functions. Pinned article: Introduction to the OurBigBook Project
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