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The key idea, is that we can't define a measure for the power set of R. Rather, we must select a large measurable subset, and the Borel sigma algebra is a good choice that matches intuitions.

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Fourier series by

Ciro Santilli 37 Updated 2025-07-16

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Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in

L^{2}

, where it always exists and converges pointwise to the function: Carleson's theorem.

Video 1.

But what is a Fourier series? by 3Blue1Brown (2019)

Source. Amazing 2D visualization of the decomposition of complex functions.

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Applications of the Fourier series by

Ciro Santilli 37 Updated 2025-07-16

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Solving partial differential equations with the Fourier series by

Ciro Santilli 37 Updated 2025-07-16

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See: 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!

Other basis besides the Fourier series show up for other equations, e.g.:

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Discrete Fourier transform by

Ciro Santilli 37 Updated 2025-07-16

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Input: a sequence of

N

complex numbers

x_{k}

Output: another sequence of

N

complex numbers

X_{k}

such that:

x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}

(1)

Intuitively, this means that we are braking up the complex signal into

N

sinusoidal frequencies:

$X_{0}$ : is kind of magic and ends up being a constant added to the signal because $e^{i 2 π \frac{kn}{N}} = e^{0} = 1$
$X_{1}$ : sinusoidal that completes one cycle over the signal. The larger the $N$ , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
$X_{2}$ : sinusoidal that completes two cycles over the signal
...
$X_{N - 1}$ : sinusoidal that completes $N - 1$ cycles over the signal

and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:

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

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.

Sample software implementations:

numpy.fft, notably see the example: numpy/fft.py

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ 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.

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Discrete Fourier transform of a real signal by

Ciro Santilli 37 Updated 2025-07-16

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See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal

The transform still has complex numbers.

Summary:

$X_{0}$ is real
$X_{1} = \overset{ˉ}{X_{N - 1}}$
$X_{2} = \overset{ˉ}{X_{N - 2}}$
$X_{k} = \overset{ˉ}{X_{N - k}}$

Therefore, we only need about half of

X_{k}

to represent the signal, as the other half can be derived by conjugation.

"Representation in terms of sines and cosines" from www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of

X_{k}

NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ with 25 points
. Source at: numpy/fft_plot.py. This plot illustrates how the DFT of a real signal is symmetric around the middle point, and so only half of the transform points are needed to reconstruct the original signal. We also see how the phase of the sinusoids determines if their DFT components are real or imaginary.

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Fast Fourier transform by

Ciro Santilli 37 Updated 2025-07-16

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An efficient algorithm to calculate the discrete Fourier transform.

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Git tips / It's not a tree, it's actually a DAG by

Ciro Santilli 37 Updated 2025-07-16

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Every tree is a directed acyclic graph.

But not every directed acyclic graph is a tree.

Example of a tree (and therefore also a DAG):

5
|
4 7
| |
3 6
|/
2
|
1

Convention in this presentation: arrows implicitly point up, just like in a git log, i.e.:

1 is parent of 2
2 is parent of 3 and 6
3 is parent of 4

and so on.

Example of a DAG that is not a tree:

7
|\
4 6
| |
3 5
|/
2
|
1

This is not a tree because there are two ways to reach 7:

2, 3, 4, 7
2, 5, 6, 7

But we often say "tree" intead of "DAG" in the context of Git because DAG sounds ugly.

Example of a graph that is not a DAG:

6
^
|
3->4
^  |
|  v
2<-5
^
|
1

This one is not acyclic because there is a cycle 2, 3, 4, 5, 2.

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Fourier transform by

Ciro Santilli 37 Updated 2025-07-16

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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.

Of course, every function defined on a finite line segment (i.e. a compact space).

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.

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Multidimensional Fourier transform by

Ciro Santilli 37 Updated 2025-07-16

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Lecture notes:

www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf Lecture 2: 2D Fourier transforms and applications by A. Zisserman (2014)

Video 1.

How the 2D FFT works by Mike X Cohen (2017)

Source. Animations showing how the 2D Fourier transform looks like for simple inpuf functions.

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Fourier inversion theorem by

Ciro Santilli 37 Updated 2025-07-16

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A set of theorems that prove under different conditions that the Fourier transform has an inverse for a given space, examples:

Carleson's theorem for $L^{2}$

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Laplace transform by

Ciro Santilli 37 Updated 2025-07-16

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Video 1.

The Laplace Transform: A Generalized Fourier Transform by Steve Brunton (2020)

Source. Explains how the Laplace transform works for functions that do not go to zero on infinity, which is a requirement for the Fourier transform. No applications in that video yet unfortunately.

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History of the Fourier series by

Ciro Santilli 37 Updated 2025-07-16

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First published by Fourier in 1807 to solve the heat equation.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade