Editing Fourier transform (section)

== Definition ==

The Fourier transform of a complex-valued (Lebesgue) integrable function <math>f(x)</math> on the real line, is the complex valued function <math>\hat{f}(\xi)</math>, defined by the integral{{sfn|Pinsky|2002|p=91}}
{{Equation box 1|title =Fourier transform
|indent =:|cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = {{NumBlk||
<math>\widehat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-i 2\pi \xi x}\,dx, \quad \forall \xi \in \mathbb{R}.</math> &nbsp; &nbsp;
|{{EquationRef|Eq.1}}}}
}}
Evaluating the Fourier transform for all values of <math>\xi</math> produces the ''frequency-domain'' function, and it converges at all frequencies to a continuous function tending to zero at infinity. If <math>f(x)</math> decays with all derivatives, i.e.,
<math display="block">\lim_{|x|\to\infty} f^{(n)}(x) = 0, \quad \forall n\in \mathbb{N},</math> then <math>\widehat f</math> converges for all frequencies and, by the [[Riemann–Lebesgue lemma]], <math>\widehat f</math> also decays with all derivatives.

First introduced in [[Joseph Fourier|Fourier's]] ''Analytical Theory of Heat''.,<ref>{{harvnb|Fourier|1822|p=525}}</ref><ref>{{harvnb|Fourier|1878|p=408}}</ref><ref>{{harvtxt|Jordan|1883}} proves on pp.&nbsp;216–226 the [[Fourier inversion theorem#Fourier integral theorem|Fourier integral theorem]] before studying Fourier series.</ref><ref>{{harvnb|Titchmarsh|1986|p=1}}</ref> the corresponding inversion formula for "[[Fourier inversion theorem#Conditions on the function|sufficiently nice]]" functions is given by the [[Fourier inversion theorem]], i.e., 
{{Equation box 1|title = Inverse transform
|indent =:|cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA
|equation = {{NumBlk||
<math>f(x) = \int_{-\infty}^{\infty} \widehat f(\xi)\ e^{i 2 \pi \xi x}\,d\xi,\quad \forall\ x \in \mathbb R.</math> &nbsp; &nbsp;
|{{EquationRef|Eq.2}}}}
}}
The functions <math>f</math> and <math>\widehat{f}</math> are referred to as a '''Fourier transform pair'''.<ref>{{harvnb|Rahman|2011|p=10}}.</ref>&nbsp; A common notation for designating transform pairs is''':'''<ref>{{harvnb|Oppenheim|Schafer|Buck|1999|p=58}}</ref>
<math display="block">f(x)\ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \widehat f(\xi),</math> &nbsp; for example &nbsp; <math>\operatorname{rect}(x)\ \stackrel{\mathcal{F}}{\longleftrightarrow}\  \operatorname{sinc}(\xi).</math>

By analogy, the [[Fourier series]] can be regarded as an abstract Fourier transform on the group <math>\mathbb{Z}</math> of [[integers]]. That is, the '''synthesis''' of a sequence of complex numbers <math>c_n</math> is defined by the Fourier transform
<math display="block">f(x) = \sum_{n=-\infty}^\infty c_n\, e^{i 2\pi \tfrac{n}{P}x},</math>
such that <math>c_n</math> are given by the inversion formula, i.e., the '''analysis'''
<math display="block">c_n =  \frac{1}{P} \int_{-P/2}^{P/2} f(x) \, e^{-i 2\pi \frac{n}{P}x} \, dx,</math>
for some complex-valued, <math>P</math>-periodic function <math>f(x)</math> defined on a bounded interval <math>[-P/2, P/2] \in \mathbb{R}</math>. When <math>P\to\infty,</math> the constituent [[frequency|frequencies]] are a continuum''':''' <math>\tfrac{n}{P} \to \xi \in \mathbb R,</math><ref>{{harvnb|Khare|Butola|Rajora|2023|pp=13–14}}</ref><ref>{{harvnb|Kaiser|1994|p=29}}</ref><ref>{{harvnb|Rahman|2011|p=11}}</ref>  and <math>c_n \to \hat{f}(\xi)\in\mathbb{C}</math>.<ref>{{harvnb|Dym|McKean|1985}}</ref>
 
In other words, on the finite interval <math>[-P/2, P/2]</math> the function <math>f(x)</math> has a discrete decomposition in the periodic functions <math>e^{i2\pi x n/P}</math>. On the infinite interval <math>(-\infty,\infty)</math> the function <math>f(x)</math> has a continuous decomposition in periodic functions <math>e^{i2\pi x \xi}</math>.

=== Lebesgue integrable functions ===
{{see also|Lp space#Lp spaces and Lebesgue integrals}}

A [[measurable function]] <math>f:\mathbb R\to\mathbb C</math> is called (Lebesgue) integrable if the [[Lebesgue integral]] of its absolute value is finite:
<math display="block">\|f\|_1 = \int_{\mathbb R}|f(x)|\,dx < \infty.</math>
If <math>f</math> is Lebesgue integrable then the Fourier transform, given by {{EquationNote|Eq.1}}, is well-defined for all <math>\xi\in\mathbb R</math>.{{sfn|Stade|2005|pp=298-299}} Furthermore, <math>\widehat f\in L^\infty\cap C(\mathbb R)</math> is bounded, [[uniformly continuous]] and (by the [[Riemann–Lebesgue lemma]]) zero at infinity.

The space <math>L^1(\mathbb R)</math> is the space of measurable functions for which the norm <math>\|f\|_1</math> is finite, modulo the [[Equivalence class|equivalence relation]] of equality [[almost everywhere]].  The Fourier transform on <math>L^1(\mathbb R)</math> is [[Bijection, injection and surjection|one-to-one]].  However, there is no easy characterization of the image, and thus no easy characterization of the inverse transform.  In particular, {{EquationNote|Eq.2}} is no longer valid, as it was stated only under the hypothesis that <math>f(x)</math> decayed with all derivatives.

While {{EquationNote|Eq.1}} defines the Fourier transform for (complex-valued) functions in <math>L^1(\mathbb R)</math>, it is not well-defined for other integrability classes, most importantly the space of [[square-integrable function]]s <math>L^2(\mathbb R)</math>. For example, the function <math>f(x)=(1+x^2)^{-1/2}</math> is in <math>L^2</math> but not <math>L^1</math> and therefore the Lebesgue integral {{EquationNote|Eq.1}} does not exist. However, the Fourier transform on the dense subspace <math>L^1\cap L^2(\mathbb R) \subset L^2(\mathbb R)</math> admits a unique continuous extension to a [[unitary operator]] on <math>L^2(\mathbb R)</math>. This extension is important in part because, unlike the case of <math>L^1</math>, the Fourier transform is an [[automorphism]] of the space <math>L^2(\mathbb R)</math>.

In such cases, the Fourier transform can be obtained explicitly by [[Regularization (mathematics)|regularizing]] the integral, and then passing to a limit. In practice, the integral is often regarded as an [[improper integral]] instead of a proper Lebesgue integral, but sometimes for convergence one needs to use [[weak limit]] or [[Cauchy principal value|principal value]] instead of the (pointwise) limits implicit in an improper integral. {{harvtxt|Titchmarsh|1986}} and {{harvtxt|Dym|McKean|1985}} each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.  A general principle in working with the <math>L^2</math> Fourier transform is that Gaussians are dense in <math>L^1\cap L^2</math>, and the various features of the Fourier transform, such as its unitarity, are easily inferred for Gaussians.  Many of the  properties of the Fourier transform, can then be proven from two facts about Gaussians:{{sfn|Howe|1980}}
* that <math>e^{-\pi x^2}</math> is its own Fourier transform; and
* that the Gaussian integral <math>\int_{-\infty}^\infty e^{-\pi x^2}\,dx = 1.</math>

A feature of the <math>L^1</math> Fourier transform is that it is a homomorphism of Banach algebras from <math>L^1</math> equipped with the convolution operation to the Banach algebra of continuous functions under the <math>L^\infty</math> (supremum) norm.  The conventions chosen in this article are those of [[harmonic analysis]], and are characterized as the unique conventions such that the Fourier transform is both [[Unitary operator|unitary]] on {{math|''L''<sup>2</sup>}} and an algebra homomorphism from {{math|''L''<sup>1</sup>}} to {{math|''L''<sup>∞</sup>}}, without renormalizing the Lebesgue measure.<ref>{{harvnb|Folland|1989}}</ref>

=== Angular frequency (''&omega;'') ===
When the independent variable (<math>x</math>) represents ''time'' (often denoted by <math>t</math>), the transform variable (<math>\xi</math>) represents [[frequency]] (often denoted by <math>f</math>). For example, if time is measured in [[second]]s, then frequency is in [[hertz]]. The Fourier transform can also be written in terms of [[angular frequency]], <math>\omega = 2\pi \xi,</math> whose units are [[radian]]s per second.

The substitution <math>\xi = \tfrac{\omega}{2 \pi}</math> into {{EquationNote|Eq.1}}  produces this convention, where function <math>\widehat f</math> is relabeled <math>\widehat {f_1}:</math>
<math display="block">\begin{align}
\widehat {f_3}(\omega) &\triangleq  \int_{-\infty}^{\infty} f(x)\cdot e^{-i\omega x}\, dx = \widehat{f_1}\left(\tfrac{\omega}{2\pi}\right),\\
f(x) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat{f_3}(\omega)\cdot e^{i\omega x}\, d\omega.
\end{align}
</math>
Unlike the {{EquationNote|Eq.1}} definition, the Fourier transform is no longer a [[unitary transformation]], and there is less symmetry between the formulas for the transform and its inverse.  Those properties are restored by splitting the <math>2 \pi</math> factor evenly between the transform and its inverse, which leads to another convention:
<math display="block">\begin{align}
\widehat{f_2}(\omega) &\triangleq \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)\cdot e^{- i\omega x}\, dx = \frac{1}{\sqrt{2\pi}}\  \ \widehat{f_1}\left(\tfrac{\omega}{2\pi}\right), \\
f(x) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \widehat{f_2}(\omega)\cdot e^{ i\omega x}\, d\omega.
\end{align}</math>
Variations of all three conventions can be created by conjugating the complex-exponential [[integral kernel|kernel]] of both the forward and the reverse transform. The signs must be opposites.

{| class="wikitable"
|+ Summary of popular forms of the Fourier transform, one-dimensional
|-
! ordinary frequency {{mvar|ξ}} (Hz)
! unitary
| <math>\begin{align}
\widehat{f_1}(\xi)\ &\triangleq\ \int_{-\infty}^{\infty} f(x)\, e^{-i 2\pi \xi x}\, dx = \sqrt{2\pi}\ \ \widehat{f_2}(2 \pi \xi) = \widehat{f_3}(2 \pi \xi) \\
f(x) &= \int_{-\infty}^{\infty} \widehat{f_1}(\xi)\, e^{i 2\pi x \xi}\, d\xi \end{align}</math>
|-
! rowspan="2" | angular frequency {{mvar|ω}} (rad/s)
! unitary
| <math>\begin{align}
\widehat{f_2}(\omega)\ &\triangleq\ \frac{1}{\sqrt{2\pi}}\ \int_{-\infty}^{\infty} f(x)\,  e^{-i \omega x}\, dx = \frac{1}{\sqrt{2\pi}}\ \ \widehat{f_1} \! \left(\frac{\omega}{2 \pi}\right) = \frac{1}{\sqrt{2\pi}}\ \ \widehat{f_3}(\omega) \\
f(x) &= \frac{1}{\sqrt{2\pi}}\ \int_{-\infty}^{\infty} \widehat{f_2}(\omega)\, e^{i \omega x}\, d\omega \end{align}</math>
|-
! non-unitary
| <math>\begin{align}
\widehat{f_3}(\omega) \ &\triangleq\ \int_{-\infty}^{\infty} f(x)\,  e^{-i\omega x}\, dx = \widehat{f_1} \left(\frac{\omega}{2 \pi}\right) = \sqrt{2\pi}\ \ \widehat{f_2}(\omega) \\
f(x) &= \frac{1}{2 \pi} \int_{-\infty}^{\infty} \widehat{f_3}(\omega)\, e^{i \omega x}\, d\omega \end{align}</math>
|}

{| class="wikitable"
|+ Generalization for {{math|''n''}}-dimensional functions
|-
! ordinary frequency {{mvar|ξ}} (Hz)
! unitary
| <math>\begin{align}
\widehat{f_1}(\xi)\ &\triangleq\ \int_{\mathbb{R}^n} f(x) e^{-i 2\pi \xi\cdot x}\, dx = (2 \pi)^\frac{n}{2}\widehat{f_2}(2\pi \xi) = \widehat{f_3}(2\pi \xi) \\
f(x) &= \int_{\mathbb{R}^n} \widehat{f_1}(\xi) e^{i 2\pi \xi\cdot x}\, d\xi \end{align}</math>
|-
! rowspan="2" | angular frequency {{mvar|ω}} (rad/s)
! unitary
| <math>\begin{align}
\widehat{f_2}(\omega)\ &\triangleq\ \frac{1}{(2 \pi)^\frac{n}{2}} \int_{\mathbb{R}^n} f(x) e^{-i \omega\cdot x}\, dx = \frac{1}{(2 \pi)^\frac{n}{2}} \widehat{f_1} \! \left(\frac{\omega}{2 \pi}\right) = \frac{1}{(2 \pi)^\frac{n}{2}} \widehat{f_3}(\omega) \\
f(x) &= \frac{1}{(2 \pi)^\frac{n}{2}} \int_{\mathbb{R}^n} \widehat{f_2}(\omega)e^{i \omega\cdot x}\, d\omega \end{align}</math>
|-
! non-unitary
| <math>\begin{align}
\widehat{f_3}(\omega) \ &\triangleq\ \int_{\mathbb{R}^n} f(x) e^{-i\omega\cdot x}\, dx = \widehat{f_1} \left(\frac{\omega}{2 \pi}\right) = (2 \pi)^\frac{n}{2} \widehat{f_2}(\omega) \\
f(x) &= \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{f_3}(\omega) e^{i \omega\cdot x}\, d\omega \end{align}</math>
|}