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:
~8GB,
lsblk reports 7796176 * 1KB = 7983284224 bytes.They are shaped like bicornes, which is super cool, but also super impractical!
Markings: "AX ÉCOLE POLYTECHNIQUE PROMOTION X2009"
From Ubuntu 20.04 on an ext4 formatted one:With Linux Unified Key Setup + ext4 the results are similar, maybe hdparam bypasses it?
/dev/sdb:
Timing cached reads: 28656 MB in 1.99 seconds = 14421.31 MB/sec
SG_IO: bad/missing sense data, sb[]: 70 00 05 00 00 00 00 0a 00 00 00 00 20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
Timing buffered disk reads: 42 MB in 3.03 seconds = 13.88 MB/sec/dev/sdb:
Timing cached reads: 28326 MB in 1.99 seconds = 14251.55 MB/sec
SG_IO: bad/missing sense data, sb[]: 70 00 05 00 00 00 00 0a 00 00 00 00 20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
Timing buffered disk reads: 38 MB in 3.11 seconds = 12.23 MB/secgnome-disks LUKS + ext4 benchmark with default params also gives about 14 MB/s.Riesz-Fischer theorem is a norm version of it, and Carleson's theorem is stronger pointwise almost everywhere version.
Note that the Riesz-Fischer theorem is weaker because the pointwise limit could not exist just according to it: norm sequence convergence does not imply pointwise convergence.
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.
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!
Output: another sequence of complex numbers such that:Intuitively, this means that we are braking up the complex signal into sinusoidal frequencies:and is the amplitude of each sine.
- : is kind of magic and ends up being a constant added to the signal because
- : sinusoidal that completes one cycle over the signal. The larger the , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
- : sinusoidal that completes two cycles over the signal
- ...
- : sinusoidal that completes cycles over the signal
Motivation: similar to the Fourier transform:In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.
- compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
- noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise
Sample software implementations:
- numpy.fft, notably see the example: numpy/fft.py
DFT of with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point.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.
A set of theorems that prove under different conditions that the Fourier transform has an inverse for a given space, examples:
Previously, updates were being done with more focus to sponsors in the format of the child sections to this section. That format is now retired in favor of the more direct Section "Updates" format.
First published by Fourier in 1807 to solve the heat equation.
Just by havin the notion of neighbourhood, concepts such as limit and continuity can be defined without the need to specify a precise numerical value to the distance between two points with a metric.
As an example. consider the orthogonal group, which is also naturally a topological space. That group does not usually have a notion of distance defined for it by default. However, we can still talk about certain properties of it, e.g. that the orthogonal group is compact, and that the orthogonal group has two connected components.
Pinned article: Introduction to the OurBigBook Project
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Intro to OurBigBook
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Figure 3. Visual Studio Code extension installation.
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