F(f(at))(ω)=F(f)(ω/a)/|a|

$ \forall a\in\R\setminus\{0\}:\mathcal F(f(at))(\omega)=\frac1{|a|}\mathcal F(f)\left(\frac\omega a\right)

証明

$ \forall a\in\R\setminus\{0\}:

$ \mathcal F(f(at))(\omega)=\frac1{\sqrt{2\pi}}\int_\R f(at)e^{-i\omega t}\mathrm dt

$ =\frac1a\frac1{\sqrt{2\pi}}\int_{t\in\R} f^*(at)e^{-i\omega at\frac1a}\mathrm d(at)

$ =\begin{dcases}\frac1a\frac1{\sqrt{2\pi}}\int_\R f^*(t)e^{-i\frac {\omega}at}\mathrm dt&\text{if }a>0\\\frac1a\frac1{\sqrt{2\pi}}\int_{-\R} f^*(t)e^{-i\frac {\omega}at}\mathrm dt&\text{if }a<0\end{dcases}

$ = \frac1{|a|}\mathcal F(f)\left(\frac\omega a\right)

References