卷积定理

（重定向自摺積定理

${\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}$

${\displaystyle {\mathcal {F}}\{f\cdot g\}={\mathcal {F}}\{f\}*{\mathcal {F}}\{g\}}$

${\displaystyle f*g={\mathcal {F}}^{-1}{\big \{}{\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}{\big \}}}$

证明

fg属于L1(Rn)。${\displaystyle F}$ ${\displaystyle f}$ 的傅里叶变换，${\displaystyle G}$ ${\displaystyle g}$ 的傅里叶变换：

${\displaystyle F(\nu )={\mathcal {F}}\{f\}=\int _{\mathbb {R} ^{n}}f(x)e^{-2\pi ix\cdot \nu }\,\mathrm {d} x}$
${\displaystyle G(\nu )={\mathcal {F}}\{g\}=\int _{\mathbb {R} ^{n}}g(x)e^{-2\pi ix\cdot \nu }\,\mathrm {d} x,}$

${\displaystyle h(z)=\int \limits _{\mathbb {R} }f(x)g(z-x)\,\mathrm {d} x.}$

${\displaystyle \int \!\!\int |f(z)g(x-z)|\,dx\,dz=\int |f(z)|\int |g(z-x)|\,dx\,dz=\int |f(z)|\,\|g\|_{1}\,dz=\|f\|_{1}\|g\|_{1}.}$

{\displaystyle {\begin{aligned}H(\nu )={\mathcal {F}}\{h\}&=\int _{\mathbb {R} }h(z)e^{-2\pi iz\cdot \nu }\,dz\\&=\int _{\mathbb {R} }\int _{\mathbb {R} ^{n}}f(x)g(z-x)\,dx\,e^{-2\pi iz\cdot \nu }\,dz.\end{aligned}}}

${\displaystyle H(\nu )=\int _{\mathbb {R} }f(x)\left(\int _{\mathbb {R} ^{n}}g(z-x)e^{-2\pi iz\cdot \nu }\,dz\right)\,dx.}$

${\displaystyle H(\nu )=\int _{\mathbb {R} }f(x)\left(\int _{\mathbb {R} }g(y)e^{-2\pi i(y+x)\cdot \nu }\,dy\right)\,dx}$
${\displaystyle =\int _{\mathbb {R} }f(x)e^{-2\pi ix\cdot \nu }\left(\int _{\mathbb {R} }g(y)e^{-2\pi iy\cdot \nu }\,dy\right)\,dx}$
${\displaystyle =\int _{\mathbb {R} }f(x)e^{-2\pi ix\cdot \nu }\,dx\int _{\mathbb {R} }g(y)e^{-2\pi iy\cdot \nu }\,dy.}$

${\displaystyle H(\nu )=F(\nu )\cdot G(\nu ),}$