Fourier級数・Fourier変換の係数倍のvariation
A
$ \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
B
$ \mathcal F(f)(\omega n):=\frac1T\int_0^Tf(t)e^{-i\omega nt}\mathrm dt
$ \mathcal F^{-1}(f)(t):=\sum_{n\in\Z}f(\omega n)e^{i\omega nt}
$ f(x)\sim\sum_{n\in\Z}c_ne^{inx}, $ c_\bullet:\Z\ni n\mapsto\frac1{2\pi}\int_Tf(x)e^{-inx}\mathrm dx\in\Complex
non-unitary:
$ \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
unitary:
$ \mathcal F(f)(\omega):=\frac1{\sqrt{2\pi}}\int_\R f(t)e^{-i\omega t}\mathrm dt
$ \mathcal F^{-1}(f)(t):=\frac1{\sqrt{2\pi}}\int_\R f(\omega)e^{i\omega t}\mathrm d\omega