フーリエ変換の諸定理
1. 言葉の定義
$ \mathcal F(f)(\omega):=\int_\R f(t)e^{-i\omega t}\mathrm dt
$ \mathcal F^{-1}(f)(t):=\frac1{2\pi}\int_\R f(\omega)e^{i\omega t}\mathrm d\omega
$ \theta(\omega):=\arg\mathcal F(f)(\omega)をFourier位相と呼ぶ $ t_\text{gr}(\omega):=\theta'(\omega)を群遅延時間と呼ぶ 2. フーリエ変換の諸定理
$ \mathcal F(t\mapsto f(t-\tau))(\omega)=\mathcal F(f)(\omega)e^{-i\omega\tau}
2. 合成積のFourier変換$ {\cal F}(x*y)(\omega)=\sqrt{2\pi}{\cal F}(x)(\omega){\cal F}(y)(\omega) 4. $ \int_\R|f(t)|^2\mathrm dt=\int_\R|\mathcal F(f)(\omega)|^2\mathrm d\omega
$ \int_\R|f(t)|^2\mathrm dt=\int_\R f(t)\mathcal F^{-1}(\mathcal F(f))(t)^*\mathrm dt
$ =\int_\R f(t)\mathcal F^{-1}(\mathcal F(f)^*)(-t)\mathrm dt
$ =\int_\R f(t)\frac1{\sqrt{2\pi}}\int_\R\mathcal F(f)^*(\omega)e^{-i\omega t}\mathrm d\omega\mathrm dt
$ =\int_\R\mathcal F(f)^*(\omega)\frac1{\sqrt{2\pi}}\int_\R f(t)e^{-i\omega t}\mathrm dt\mathrm d\omega
$ =\int_\R\mathcal F(f)^*(\omega)\mathcal F(f)(\omega)\mathrm d\omega
$ = \int_\R|\mathcal F(f)(\omega)|^2\mathrm d\omega
6. 相互相関函数のFourier変換$ \mathcal F(f\star g)(\omega)=\mathcal F(f)(\omega)^*\mathcal F(g)(\omega) $ \mathcal F(f\star f)(\omega)=\mathcal F(f)(\omega)^*\mathcal F(f)(\omega)=|\mathcal F(f)(\omega)|^2