Fourier inversion theorem

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..

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

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)

- 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.