Laplace operator Updated 2025-07-16
Can be denoted either by:
Our default symbol is going to be:
Evil Updated 2025-07-16
Things that are not nice such as:
Set function Updated 2025-07-16
This section is about functions that operates on arbitrary sets.
Numeric function Updated 2025-07-16
This section is about functions that operate on numbers such as the integers or real numbers.
Discrete Fourier transform Updated 2025-07-16
Input: a sequence of complex numbers .
Output: another sequence of complex numbers such that:
Intuitively, this means that we are braking up the complex signal into sinusoidal frequencies:
  • : 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
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:
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:
Figure 1.
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.
Fourier transform Updated 2025-07-16
Continuous version of the Fourier series.
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.
History of the Fourier series Updated 2025-07-16
First published by Fourier in 1807 to solve the heat equation.
Axiom Updated 2025-07-16
Conjecture Updated 2025-07-16
A conjecture is an open problem in mathematics for which some famous dude gave heuristic arguments which indicate if the theorem is true or false.
Theorem Updated 2025-07-16

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