Fourier-Transformation/Faltungssatz/Fakt/Beweis

Beweis

Nach Fakt (für Dichten) angewendet auf die Addition und Fakt ist

{}{\begin{aligned}{\widehat {f*g}}({\mathfrak {u}})&={\frac {1}{(2\pi )^{n/2}}}\cdot \int _{\mathbb {R} ^{n}}e^{-{\mathrm {i} }\left\langle {\mathfrak {u}},{\mathfrak {t}}\right\rangle }(f*g)({\mathfrak {u}})d{\mathfrak {t}}\\&={\frac {1}{(2\pi )^{n/2}}}\cdot \int _{\mathbb {R} ^{n}\times \mathbb {R} ^{n}}e^{-{\mathrm {i} }\left\langle {\mathfrak {u}},{\mathfrak {t}}_{1}+{\mathfrak {t}}_{2}\right\rangle }f({\mathfrak {t}}_{1})g({\mathfrak {t}}_{2})d{\mathfrak {t}}_{1}d{\mathfrak {t}}_{2}\\&={\frac {1}{(2\pi )^{n/2}}}\cdot {\left(\int _{\mathbb {R} ^{n}}e^{-{\mathrm {i} }\left\langle {\mathfrak {u}},{\mathfrak {t}}_{1}\right\rangle }f({\mathfrak {t}}_{1})d{\mathfrak {t}}_{1}\right)}\cdot {\left(\int _{\mathbb {R} ^{n}}e^{-{\mathrm {i} }\left\langle {\mathfrak {u}},{\mathfrak {t}}_{2}\right\rangle }g({\mathfrak {t}}_{2})d{\mathfrak {t}}_{2}\right)}\\&=(2\pi )^{n/2}\cdot {\hat {f}}({\mathfrak {u}})\cdot {\hat {g}}({\mathfrak {u}}).\end{aligned}}