= Fourier transform
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Continuous version of the <Fourier series>.
Can be used to represent functions that are not periodic: https://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: <solving partial differential equations with the Fourier series>{full}, the <Fourier transform> is what you need to solve the problem when the <domain (function)> is the entire <real line>.