nLab Fourier inversion theorem

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

Proposition

(Fourier inversion theorem)

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

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

given by

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

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

(e.g. Hörmander, theorem 7.1.5)

Related theorems

• Parseval's theorem

• Paley-Wiener-Schwartz theorem

References

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