Editing Fourier inversion theorem (section)

===Tempered distributions===
{{see also|Fourier transform#Tempered distributions}}
The Fourier transform may be defined on the space of [[tempered distribution]]s <math>\mathcal{S}'(\mathbb{R}^n)</math> by duality of the Fourier transform on the space of Schwartz functions. Specifically for <math>f\in\mathcal{S}'(\mathbb{R}^n)</math> and for all test functions <math>\varphi\in\mathcal S(\mathbb{R}^n)</math> we set
:<math>\langle \mathcal{F}f,\varphi\rangle := \langle f,\mathcal{F}\varphi\rangle,</math>
where <math>\mathcal{F}\varphi</math> is defined using the integral formula.{{sfn|Folland|1992|p=333}} If <math>f \in L^1(\mathbb R^n) \cap L^2(\mathbb R^n)</math> then this agrees with the usual definition. We may define the inverse transform <math>\mathcal{F}^{-1}\colon\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)</math>, either by duality from the inverse transform on Schwartz functions in the same way, or by defining it in terms of the flip operator (where the flip operator is defined by duality). We then have

:<math>\mathcal{F}\mathcal{F}^{-1} = \mathcal{F}^{-1}\mathcal{F} = \operatorname{Id}_{\mathcal{S}'(\mathbb{R}^n)}.</math>