nLab
Fourier inversion theorem

Contents

Contents

Idea

In harmonic analysis, the Fourier inversion theorem states that the Fourier transform is an isomorphism on the Schwartz space of functions with rapidly decreasing partial derivatives. Moreover, it is its own inverse up to a prefactor and reflection at the origin.

For Fourier transform over Cartesian spaces, see e.g. Hörmander 90 ,theorem 7.1.5, theorem 7.1.10, this prop..

Statement

Let nn \in \mathbb{N} and consider n\mathbb{R}^n the Cartesian space of dimension nn.

Proposition

(Fourier inversion theorem)

The Fourier transform ()^\widehat{(-)} (def. ) on the Schwartz space 𝒮( n)\mathcal{S}(\mathbb{R}^n) (def. ) is an isomorphism, with inverse function the inverse Fourier transform

()ˇ:𝒮( n)𝒮( n) \widecheck {(-)} \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathcal{R}^n)

given by

gˇ(x)k ng(k)e 2πikxd nk(2π) n. \widecheck g (x) \;\coloneqq\; \underset{k \in \mathbb{R}^n}{\int} g(k) e^{2 \pi i k \cdot x} \, \frac{d^n k}{(2\pi)^n} \,.

Hence in the language of harmonic analysis the function gˇ: n\widecheck g \colon \mathbb{R}^n \to \mathbb{C} is the superposition of plane waves in which the plane wave with wave vector k nk\in \mathbb{R}^n appears with amplitude g(k)g(k).

(e.g. Hörmander, theorem 7.1.5)

References

  • Lars Hörmander, theorem 7.1.5 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Last revised on November 7, 2017 at 11:45:45. See the history of this page for a list of all contributions to it.