窓函数のFourier変換

$ \mathcal F(t\mapsto\llbracket a\le t\le b\rrbracket)(\omega)=\frac{b}{\sqrt{2\pi}}\operatorname{sinc}\left(\frac{\omega b}{2}\right)e^{-i\frac{\omega b}{2}}-\frac{a}{\sqrt{2\pi}}\operatorname{sinc}\left(-\frac{\omega a}{2}\right)e^{i\frac{\omega a}{2}}

$ \mathcal F(t\mapsto\llbracket0\le t\le b\rrbracket)(\omega)= \frac{1}{\sqrt{2\pi}}\int_\R\llbracket0\le t\le b\rrbracket e^{-i\omega t}\mathrm dt

$ = \frac{1}{\sqrt{2\pi}}\int_{0}^{b} e^{-i\omega t}\mathrm dt

$ =-\frac1{i\omega}\frac{1}{\sqrt{2\pi}}\int_{t=0}^{t=b}\mathrm d\left(e^{-i\omega t}\right)

$ =-\frac1{i\omega}\frac{1}{\sqrt{2\pi}}(e^{-i\omega b}-1)

$ =\frac{1}{\sqrt{2\pi}}\frac{e^{i\frac{\omega b}{2}}-e^{-i\frac{\omega b}{2}}}{2i}\frac{e^{-i\frac{\omega b}{2}}}{\omega}

$ =\frac{b}{\sqrt{2\pi}}\frac{\sin\left(\frac{\omega b}{2}\right)}{\frac{\omega b}{2}}e^{-i\frac{\omega b}{2}}

$ =\frac{b}{\sqrt{2\pi}}\operatorname{sinc}\left(\frac{\omega b}{2}\right)e^{-i\frac{\omega b}{2}}

$ \operatorname{sinc}\theta:=\frac{\sin\theta}{\theta}

$ \mathcal F(t\mapsto\llbracket a\le t\le 0\rrbracket)(\omega)=\mathcal F(t\mapsto\llbracket 0\le-t\le-a\rrbracket)(\omega)

$ = \mathcal F(t\mapsto\llbracket0\le t\le-a\rrbracket)(\omega)

$ =-\frac{a}{\sqrt{2\pi}}\operatorname{sinc}\left(-\frac{\omega a}{2}\right)e^{i\frac{\omega a}{2}}

$ \therefore\mathcal F(t\mapsto\llbracket a\le t\le b\rrbracket)(\omega)=\frac{b}{\sqrt{2\pi}}\operatorname{sinc}\left(\frac{\omega b}{2}\right)e^{-i\frac{\omega b}{2}}-\frac{a}{\sqrt{2\pi}}\operatorname{sinc}\left(-\frac{\omega a}{2}\right)e^{i\frac{\omega a}{2}}

$ a\le t\le b

$ \iff \frac{a-b}2\le t-\frac{a+b}2\le-\frac{a-b}2

$ \iff\left|t-\frac{b+a}2\right|\le\frac{b-a}2

$ \iff\left|\frac t{b-a}-\frac12\frac{b+a}{b-a}\right|\le\frac12

$ \therefore\llbracket a\le t\le b\rrbracket=\sqcap\left(\frac t{b-a}-\frac12\frac{b+a}{b-a}\right)

ただし、不連続点の値を除く

$ \mathcal F(t\mapsto\llbracket a\le t\le b\rrbracket)(\omega)=|b-a|e^{-i\omega\frac12\frac{b+a}{b-a}}\mathcal F(\sqcap)\left((b-a)\omega\right)

$ = (b-a)e^{-i\omega\frac12\frac{b+a}{b-a}}\frac1{\sqrt{2\pi}}\operatorname{sinc}\left(\frac{b-a}2\omega\right)

$ \because a<b

$ \mathcal F(t\mapsto\llbracket0\le t\le b\rrbracket)(\omega)=\frac b{\sqrt{2\pi}}\operatorname{sinc}\left(\frac{\omega b}{2}\right)e^{-\frac12i\omega}