Editing Fourier transform (section)

== Fourier transform on function spaces ==
{{see also|Riesz–Thorin theorem}}
The definition of the Fourier transform naturally extends from <math>L^1(\mathbb R)</math> to <math>L^1(\mathbb R^n)</math>. That is, if <math>f \in L^1(\mathbb{R}^n)</math> then the Fourier transform 
<math>\mathcal{F}:L^1(\mathbb{R}^n) \to L^\infty(\mathbb{R}^n)</math> is given by <math display="block">f(x)\mapsto \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x)e^{-i 2\pi \xi\cdot x}\,dx, \quad \forall \xi \in \mathbb{R}^n.</math>
This operator is [[bounded operator|bounded]] as
<math display="block">\sup_{\xi \in \mathbb{R}^n}\left\vert\hat{f}(\xi)\right\vert \leq \int_{\mathbb{R}^n} \vert f(x)\vert \,dx,</math>
which shows that its [[operator norm]] is bounded by {{math|1}}. The [[Riemann–Lebesgue lemma]] shows that if <math>f\in L^1(\mathbb{R}^n)</math> then its Fourier transform actually belongs to the [[Function space#Functional analysis|space of continuous functions which vanish at infinity]], i.e., <math>\hat{f} \in C_{0}(\mathbb{R}^n)\subset L^{\infty}(\mathbb{R}^n)</math>.{{sfn|Stein|Weiss|1971|pp=1-2}} Furthermore, the [[Image (mathematics)|image]] of <math>L^1</math> under <math>\mathcal{F}</math> is a strict subset of <math>C_{0}(\mathbb{R}^n)</math>.
 
Similarly to the case of one variable, the Fourier transform can be defined on <math>L^2(\mathbb R^n)</math>. The Fourier transform in <math>L^2(\mathbb R^n)</math> is no longer given by an ordinary Lebesgue integral, although it can be computed by an [[improper integral]], i.e.,
<math display="block">\hat{f}(\xi) = \lim_{R\to\infty}\int_{|x|\le R} f(x) e^{-i 2\pi\xi\cdot x}\,dx</math>
where the limit is taken in the {{math|''L''<sup>2</sup>}} sense.<ref>More generally, one can take a sequence of functions that are in the intersection of {{math|''L''<sup>1</sup>}} and {{math|''L''<sup>2</sup>}} and that converges to {{mvar|f}} in the {{math|''L''<sup>2</sup>}}-norm, and define the Fourier transform of {{mvar|f}} as the {{math|''L''<sup>2</sup>}} -limit of the Fourier transforms of these functions.</ref><ref>{{cite web|url=https://statweb.stanford.edu/~candes/teaching/math262/Lectures/Lecture03.pdf|title=Applied Fourier Analysis and Elements of Modern Signal Processing Lecture 3 |date= January 12, 2016|access-date=2019-10-11}}</ref>

Furthermore, <math>\mathcal{F}:L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)</math> is a [[unitary operator]].{{sfn|Stein|Weiss|1971|loc=Thm. 2.3}} For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any {{math|''f'', ''g'' ∈ ''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} we have
<math display="block">\int_{\mathbb{R}^n} f(x)\mathcal{F}g(x)\,dx = \int_{\mathbb{R}^n} \mathcal{F}f(x)g(x)\,dx. </math>

In particular, the image of {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} is itself under the Fourier transform.

=== On other ''L''<sup>''p''</sup> ===
For <math>1<p<2</math>, the Fourier transform can be defined on <math>L^p(\mathbb R)</math> by [[Marcinkiewicz interpolation]], which amounts to decomposing such functions into a fat tail part in {{math|''L''<sup>2</sup>}} plus a fat body part in {{math|''L''<sup>1</sup>}}. In each of these spaces, the Fourier transform of a function in {{math|''L''{{isup|''p''}}('''R'''<sup>''n''</sup>)}} is in {{math|''L''{{isup|''q''}}('''R'''<sup>''n''</sup>)}}, where {{math|1=''q'' = {{sfrac|''p''|''p'' − 1}}}} is the [[Hölder conjugate]] of {{mvar|p}} (by the [[Hausdorff–Young inequality]]). However, except for {{math|1=''p'' = 2}}, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in {{math|''L''{{isup|''p''}}}} for the range {{math|2 < ''p'' < ∞}} requires the study of distributions.{{sfn|Katznelson|2004}} In fact, it can be shown that there are functions in {{math|''L''{{isup|''p''}}}} with {{math|''p'' > 2}} so that the Fourier transform is not defined as a function.<ref name="Stein-Weiss-1971" />

=== 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.