合成積と畳み込み積分
合成積∗*∗の定義
x∗y:t↦∫Rx(τ)y(t−τ)dτx*y:t\mapsto\int_\R x(\tau)y(t-\tau)\mathrm d\taux∗y:t↦∫Rx(τ)y(t−τ)dτ
別名
畳み込み
合成積のFourier変換は、Fourier変換の積になる
F(x∗y)(ω)=F(x)(ω)F(y)(ω){\cal F}(x*y)(\omega)={\cal F}(x)(\omega){\cal F}(y)(\omega)F(x∗y)(ω)=F(x)(ω)F(y)(ω)
証明
F(x∗y)(ω)=∫(t,τ)∈R2x(τ)y(t−τ)e−iωtdτdt{\cal F}(x*y)(\omega)=\int_{(t,\tau)\in\R^2}x(\tau)y(t-\tau)e^{-i\omega t}\mathrm d\tau\mathrm dtF(x∗y)(ω)=∫(t,τ)∈R2x(τ)y(t−τ)e−iωtdτdt