Editing Fourier transform (section)

=== 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]].