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The reason why the epsilon delta definition is so venerated is that it fits directly into well known methods of the formalization of mathematics, making the notion completely precise.

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Convergent series by

Ciro Santilli 40 Updated 2025-07-16

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Plancherel theorem by

Ciro Santilli 40 Updated 2025-07-16

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Some sources say that this is just the part that says that the norm of a

L^{2}

function is the same as the norm of its Fourier transform.

Others say that this theorem actually says that the Fourier transform is bijective.

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

l^{1}

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.

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Continuous problems are simpler than discrete ones by

Ciro Santilli 40 Updated 2025-07-16

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This is a general philosophy that Ciro Santilli, and likely others, observes over and over.

Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.

Some good examples of that:

complex discrete problems:
- classification of finite groups
simple continuous problems:
- characterization of Lie groups

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The Fourier transform is a bijection in

L^{2}

Ciro Santilli 40 Updated 2025-07-16

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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.

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Solving partial differential equations with the Fourier series by

Ciro Santilli 40 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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Chain rule by

Ciro Santilli 40 Updated 2025-07-16

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Here's an example of the chain rule. Suppose we want to calculate:

\frac{d}{d x} e^{2 x}

(1)

So we have:

f (x) = e^{x} g (x) = 2 x

(2)

and so:

f^{'} (x) = e^{x} g^{'} (x) = 2

(3)

Therefore the final result is:

f^{'} (g (x)) g^{'} (x) = e^{2 x} 2 = 2 e^{2 x}

(4)

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Discrete Fourier transform by

Ciro Santilli 40 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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Normalized DFT by

Ciro Santilli 40 Updated 2025-07-16

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There are actually two possible definitions for the DFT:

1/N, given as "the default" in many sources:
$x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}$
(1)
$1/ N$ , known as the "normalized DFT" by some sources: www.dsprelated.com/freebooks/mdft/Normalized_DFT.html, definition which we adopt:
$x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{kn}{N}}$
(2)

The

1/ N

is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.

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Infinitely differentiable function by

Ciro Santilli 40 Updated 2025-07-16

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Bump function by

Ciro Santilli 40 Updated 2025-07-16

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Fourier transform by

Ciro Santilli 40 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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Maxima and minima by

Ciro Santilli 40 Updated 2025-07-16

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Given a function

f

from some space. For beginners the real numbers but more generally topological spaces should work in general
to the real numbers

we want to find the points

x

of the domain of

f

where the value of

f

is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of

x

In the case of Functionals, this problem is treated under the theory of the calculus of variations.

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Fourier inversion theorem by

Ciro Santilli 40 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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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.

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. Source.

We have two killer features:

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- 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.
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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.
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Figure 4.
Visual Studio Code extension tree navigation
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Figure 5.
Web editor
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Video 3.
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. 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
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