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 a 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.

- Fourier series | 143, 555, 8
- Calculus | 17, 7k, 159
- Mathematics | 17, 13k, 329
- Ciro Santilli's Homepage | 262, 181k, 3k

- Fourier inversion theorem | 41
- Fourier series | 143, 555, 8
- Fourier transform | 186, 321, 3
- Laplace transform | 51
- Plancherel theorem | 150, 248, 2
- Position and momentum space | 187, 435, 4
- Uncertainty principle | 292, 1k, 10