Continuous version of the Fourier series.
Can be used to represent functions that are not periodic: 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.
Lecture notes:
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.
A set of theorems that prove under different conditions that the Fourier transform has an inverse for a given space, examples:
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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