Editing Fourier transform (section)

=== Convolution theorem ===
{{Main|Convolution theorem}}

The Fourier transform translates between [[convolution]] and multiplication of functions. If {{math|''f''(''x'')}} and {{math|''g''(''x'')}} are integrable functions with Fourier transforms {{math|''f̂''(''ξ'')}} and {{math|''ĝ''(''ξ'')}} respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms {{math|''f̂''(''ξ'')}} and {{math|''ĝ''(''ξ'')}} (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if:
<math display="block">h(x) = (f*g)(x) = \int_{-\infty}^\infty f(y)g(x - y)\,dy,</math>
where {{math|∗}} denotes the convolution operation, then:
<math display="block">\hat{h}(\xi) = \hat{f}(\xi)\, \hat{g}(\xi).</math>

In [[LTI system theory|linear time invariant (LTI) system theory]], it is common to interpret {{math|''g''(''x'')}} as the [[impulse response]] of an LTI system with input {{math|''f''(''x'')}} and output {{math|''h''(''x'')}}, since substituting the [[Dirac delta function|unit impulse]] for {{math|''f''(''x'')}} yields {{math|1=''h''(''x'') = ''g''(''x'')}}. In this case, {{math|''ĝ''(''ξ'')}} represents the [[frequency response]] of the system.

Conversely, if {{math|''f''(''x'')}} can be decomposed as the product of two square integrable functions {{math|''p''(''x'')}} and {{math|''q''(''x'')}}, then the Fourier transform of {{math|''f''(''x'')}} is given by the convolution of the respective Fourier transforms {{math|''p̂''(''ξ'')}} and {{math|''q̂''(''ξ'')}}.