Editing Fourier transform (section)

== Other notations ==
Other common notations for <math>\hat f(\xi)</math> include:
<math display="block">\tilde{f}(\xi),\ F(\xi),\ \mathcal{F}\left(f\right)(\xi),\ \left(\mathcal{F}f\right)(\xi),\ \mathcal{F}(f),\ \mathcal{F}\{f\},\ \mathcal{F} \bigl(f(t)\bigr),\ \mathcal{F} \bigl\{f(t)\bigr\}.</math>

In the sciences and engineering it is also common to make substitutions like these:
<math display="block">\xi \rightarrow f, \quad x \rightarrow t, \quad f \rightarrow x,\quad \hat f \rightarrow X.  </math>

So the transform pair <math>f(x)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \hat{f}(\xi)</math> can become <math>x(t)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ X(f)</math>

A disadvantage of the capital letter notation is when expressing a transform such as <math>\widehat{f\cdot g}</math> or <math>\widehat{f'},</math> which become the more awkward <math>\mathcal{F}\{f\cdot g\}</math> and <math>\mathcal{F} \{ f' \} . </math>

In some contexts such as particle physics, the same symbol <math>f</math> may be used for both for a function as well as it Fourier transform, with the two only distinguished by their [[Argument of a function|argument]]  I.e. <math>f(k_1 + k_2)</math> would refer to the Fourier transform because of the momentum argument, while <math>f(x_0 + \pi \vec r)</math> would refer to the original function because of the positional argument. Although tildes may be used as in <math>\tilde{f}</math> to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more [[Lorentz invariant]] form, such as <math>\tilde{dk} = \frac{dk}{(2\pi)^32\omega}</math>, so care must be taken.  Similarly, <math>\hat f</math> often denotes the [[Hilbert transform]] of <math>f</math>.

The interpretation of the complex function {{math|''f̂''(''ξ'')}} may be aided by expressing it in [[polar coordinate]] form
<math display="block">\hat f(\xi) = A(\xi) e^{i\varphi(\xi)}</math>
in terms of the two real functions {{math|''A''(''ξ'')}} and {{math|''φ''(''ξ'')}} where:
<math display="block">A(\xi) = \left|\hat f(\xi)\right|,</math>
is the [[amplitude]] and
<math display="block">\varphi (\xi) = \arg \left( \hat f(\xi) \right), </math>
is the [[phase (waves)|phase]] (see [[Arg (mathematics)|arg function]]).

Then the inverse transform can be written:
<math display="block">f(x) = \int _{-\infty}^\infty A(\xi)\ e^{ i\bigl(2\pi \xi x +\varphi (\xi)\bigr)}\,d\xi,</math>
which is a recombination of all the frequency components of {{math|''f''(''x'')}}. Each component is a complex [[sinusoid]] of the form {{math|''e''<sup>2π''ixξ''</sup>}} whose amplitude is {{math|''A''(''ξ'')}} and whose initial [[phase (waves)|phase angle]] (at {{math|1=''x'' = 0}}) is {{math|''φ''(''ξ'')}}.

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted {{mathcal|F}} and {{math|{{mathcal|F}}(''f'')}} is used to denote the Fourier transform of the function {{mvar|f}}. This mapping is linear, which means that {{mathcal|F}} can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function {{math|''f''}}) can be used to write {{math|{{mathcal|F}} ''f''}} instead of {{math|{{mathcal|F}}(''f'')}}. Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value {{mvar|ξ}} for its variable, and this is denoted either as {{math|{{mathcal|F}} ''f''(''ξ'')}} or as {{math|({{mathcal|F}} ''f'')(''ξ'')}}. Notice that in the former case, it is implicitly understood that {{mathcal|F}} is applied first to {{mvar|f}} and then the resulting function is evaluated at {{mvar|ξ}}, not the other way around.

In mathematics and various applied sciences, it is often necessary to distinguish between a function {{mvar|f}} and the value of {{mvar|f}} when its variable equals {{mvar|x}}, denoted {{math|''f''(''x'')}}. This means that a notation like {{math|{{mathcal|F}}(''f''(''x''))}} formally can be interpreted as the Fourier transform of the values of {{mvar|f}} at {{mvar|x}}. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example,
<math display="block">\mathcal F\bigl( \operatorname{rect}(x) \bigr) = \operatorname{sinc}(\xi)</math>
is sometimes used to express that the Fourier transform of a [[rectangular function]] is a [[sinc function]], or
<math display="block">\mathcal F\bigl(f(x + x_0)\bigr) = \mathcal F\bigl(f(x)\bigr)\, e^{i 2\pi x_0 \xi}</math>
is used to express the shift property of the Fourier transform.

Notice, that the last example is only correct under the assumption that the transformed function is a function of {{mvar|x}}, not of {{math|''x''<sub>0</sub>}}.

As discussed above, the [[Characteristic function (probability theory)|characteristic function]] of a random variable is the same as the [[#Fourier–Stieltjes transform|Fourier–Stieltjes transform]] of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is defined
<math display="block">E\left(e^{it\cdot X}\right)=\int e^{it\cdot x} \, d\mu_X(x).</math>

As in the case of the "non-unitary angular frequency" convention above, the factor of 2{{pi}} appears in neither the normalizing constant nor the exponent. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponent.