Editing Fourier transform (section)

=== Tempered distributions ===
{{Main|Distribution (mathematics)#Tempered distributions and Fourier transform}}
{{See also|Rigged Hilbert space}}
One might consider enlarging the domain of the Fourier transform from <math>L^1 + L^2</math> by considering [[generalized function]]s, or distributions. A distribution on <math>\mathbb{R}^n</math> is a continuous linear functional on the space <math>C_{c}^{\infty}(\mathbb{R}^n)</math> of compactly supported smooth functions (i.e. [[bump function]]s), equipped with a suitable topology.  Since <math>C_{c}^{\infty}(\mathbb{R}^n)</math> is dense in <math>L^{2}(\mathbb{R}^n)</math>, the [[Plancherel theorem]] allows one to extend the definition of the Fourier transform to general functions in <math>L^{2}(\mathbb{R}^n)</math> by continuity arguments. The strategy is then to consider the action of the Fourier transform on <math>C_{c}^{\infty}(\mathbb{R}^n)</math> and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map <math>C_{c}^{\infty}(\mathbb{R}^n)</math> to <math>C_{c}^{\infty}(\mathbb{R}^n)</math>. In fact the Fourier transform of an element in <math>C_{c}^{\infty}(\mathbb{R}^n)</math> can not vanish on an open set; see the above discussion on the uncertainty principle.{{sfn|Mallat|2009|p=45}}{{sfn|Strichartz|1994|p=150}}

The Fourier transform can also be defined for [[tempered distribution]]s <math>\mathcal S'(\mathbb R^n)</math>, dual to the space of [[Schwartz function]]s  <math>\mathcal S(\mathbb R^n)</math>. A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives, hence <math>C_{c}^{\infty}(\mathbb{R}^n)\subset  \mathcal S(\mathbb R^n)</math> and:
<math display="block">\mathcal{F}: C_{c}^{\infty}(\mathbb{R}^n) \rightarrow S(\mathbb R^n) \setminus  C_{c}^{\infty}(\mathbb{R}^n).</math> The Fourier transform is an [[automorphism]] of the Schwartz space and, by duality, also an automorphism of the space of tempered distributions.<ref name="Stein-Weiss-1971" />{{sfn|Hunter|2014}} The tempered distributions include well-behaved functions of polynomial growth, distributions of compact support as well as all the integrable functions mentioned above.

For the definition of the Fourier transform of a tempered distribution, let <math>f</math> and <math>g</math> be integrable functions, and let <math>\hat{f}</math> and <math>\hat{g}</math> be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,<ref name="Stein-Weiss-1971" />
<math display="block">\int_{\mathbb{R}^n}\hat{f}(x)g(x)\,dx=\int_{\mathbb{R}^n}f(x)\hat{g}(x)\,dx.</math>

Every integrable function <math>f</math> defines (induces) a distribution <math>T_f</math> by the relation
<math display="block">T_f(\phi)=\int_{\mathbb{R}^n}f(x)\phi(x)\,dx,\quad \forall \phi\in\mathcal S(\mathbb R^n).</math>
So it makes sense to define the Fourier transform of a tempered distribution <math>T_{f}\in\mathcal S'(\mathbb R)</math> by the duality:
<math display="block">\langle \widehat T_{f}, \phi\rangle = \langle T_{f},\widehat \phi\rangle,\quad \forall \phi\in\mathcal S(\mathbb R^n).</math>
Extending this to all tempered distributions <math>T</math> gives the general definition of the Fourier transform.

Distributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.