Editing Fourier transform (section)

== Complex domain ==
The [[integral]] for the Fourier transform
<math display="block"> \hat f (\xi) = \int _{-\infty}^\infty e^{-i 2\pi \xi t} f(t) \, dt </math>
can be studied for [[complex number|complex]] values of its argument {{mvar|ξ}}. Depending on the properties of {{mvar|f}}, this might not converge off the real axis at all, or it might converge to a [[complex analysis|complex]] [[analytic function]] for all values of {{math|''ξ'' {{=}} ''σ'' + ''iτ''}}, or something in between.<ref>{{harvnb|Paley|Wiener|1934}}</ref>

The [[Paley–Wiener theorem]] says that {{mvar|f}} is smooth (i.e., {{mvar|n}}-times differentiable for all positive integers {{mvar|n}}) and compactly supported if and only if {{math|''f̂'' (''σ'' + ''iτ'')}} is a [[holomorphic function]] for which there exists a [[constant (mathematics)|constant]] {{math|''a'' > 0}} such that for any [[integer]] {{math|''n'' ≥ 0}},
<math display="block"> \left\vert \xi ^n \hat f(\xi) \right\vert \leq C e^{a\vert\tau\vert} </math>
for some constant {{mvar|C}}. (In this case, {{mvar|f}} is supported on {{math|[−''a'', ''a'']}}.) This can be expressed by saying that {{math|''f̂''}} is an [[entire function]] which is [[rapidly decreasing]] in {{mvar|σ}} (for fixed {{mvar|τ}}) and of exponential growth in {{mvar|τ}} (uniformly in {{mvar|σ}}).<ref>{{harvnb|Gelfand|Vilenkin|1964}}</ref>

(If {{mvar|f}} is not smooth, but only {{math|''L''<sup>2</sup>}}, the statement still holds provided {{math|''n'' {{=}} 0}}.<ref>{{harvnb|Kirillov|Gvishiani|1982}}</ref>) The space of such functions of a [[complex analysis|complex variable]] is called the Paley—Wiener space. This theorem has been generalised to semisimple [[Lie group]]s.<ref>{{harvnb|Clozel|Delorme|1985|pp=331–333}}</ref>

If {{mvar|f}} is supported on the half-line {{math|''t'' ≥ 0}}, then {{mvar|f}} is said to be "causal" because the [[impulse response function]] of a physically realisable [[Filter (mathematics)|filter]] must have this property, as no effect can precede its cause. [[Raymond Paley|Paley]] and Wiener showed that then {{math|''f̂''}} extends to a [[holomorphic function]] on the complex lower half-plane {{math|''τ'' < 0}} which tends to zero as {{mvar|τ}} goes to infinity.<ref>{{harvnb|de Groot|Mazur|1984|p=146}}</ref> The converse is false and it is not known how to characterise the Fourier transform of a causal function.<ref>{{harvnb|Champeney|1987|p=80}}</ref>

=== Laplace transform ===
{{See also|Laplace transform#Fourier transform}}
The Fourier transform {{math|''f̂''(''ξ'')}} is related to the [[Laplace transform]] {{math|''F''(''s'')}}, which is also used for the solution of [[differential equation]]s and the analysis of [[Filter (signal processing)|filter]]s.

It may happen that a function {{mvar|f}} for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the [[complex plane]].

For example, if {{math|''f''(''t'')}} is of exponential growth, i.e.,
<math display="block"> \vert f(t) \vert < C e^{a\vert t\vert} </math>
for some constants {{math|''C'', ''a'' ≥ 0}}, then<ref name="Kolmogorov-Fomin-1999">{{harvnb|Kolmogorov|Fomin|1999}}</ref>
<math display="block"> \hat f (i\tau) = \int _{-\infty}^\infty e^{ 2\pi \tau t} f(t) \, dt, </math>
convergent for all {{math|2π''τ'' < −''a''}}, is the [[two-sided Laplace transform]] of {{mvar|f}}.

The more usual version ("one-sided") of the Laplace transform is
<math display="block"> F(s) = \int_0^\infty f(t) e^{-st} \, dt.</math>

If {{mvar|f}} is also causal, and analytical, then: <math> \hat f(i\tau) = F(-2\pi\tau).</math> Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variable {{math|''s'' {{=}} ''i''2π''ξ''}}.

From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements. Two particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle. Such designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb.

Furthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response. There, only the delay circuit in-between admits a classical Fourier description, which is critical. Both the circuits to the side are unstable, and do not admit a convergent Fourier decomposition. However, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel.

In modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea of [[harmonic analysis]].

=== Inversion ===
Still with <math>\xi = \sigma+ i\tau</math>, if <math>\widehat f</math> is complex analytic for {{math|''a'' ≤ ''τ'' ≤ ''b''}}, then

<math display="block"> \int _{-\infty}^\infty \hat f (\sigma + ia) e^{ i 2\pi \xi t} \, d\sigma = \int _{-\infty}^\infty \hat f (\sigma + ib) e^{ i 2\pi \xi t} \, d\sigma </math>
by [[Cauchy's integral theorem]]. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.<ref>{{harvnb|Wiener|1949}}</ref>

Theorem: If {{math|1=''f''(''t'') = 0}} for {{math|''t'' < 0}}, and {{math|{{abs|''f''(''t'')}} < ''Ce''<sup>''a''{{abs|''t''}}</sup>}} for some constants {{math|''C'', ''a'' > 0}}, then
<math display="block"> f(t) = \int_{-\infty}^\infty \hat f(\sigma + i\tau) e^{i 2 \pi \xi t} \, d\sigma,</math>
for any {{math|''τ'' < −{{sfrac|''a''|2π}}}}.

This theorem implies the [[inverse Laplace transform#Mellin's inverse formula|Mellin inversion formula]] for the Laplace transformation,<ref name="Kolmogorov-Fomin-1999" />
<math display="block"> f(t) = \frac 1 {i 2\pi} \int_{b-i\infty}^{b+i\infty} F(s) e^{st}\, ds</math>
for any {{math|''b'' > ''a''}}, where {{math|''F''(''s'')}} is the Laplace transform of {{math|''f''(''t'')}}.

The hypotheses can be weakened, as in the results of Carleson and Hunt, to {{math|''f''(''t'') ''e''<sup>−''at''</sup>}} being {{math|''L''<sup>1</sup>}}, provided that {{mvar|f}} be of bounded variation in a closed neighborhood of {{mvar|t}} (cf. [[Dini test]]), the value of {{mvar|f}} at {{mvar|t}} be taken to be the [[arithmetic mean]] of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.<ref>{{harvnb|Champeney|1987|p=63}}</ref>

{{math|''L''<sup>2</sup>}} versions of these inversion formulas are also available.<ref>{{harvnb|Widder|Wiener|1938|p=537}}</ref>