Fourier-Transformation/Translationseigenschaften/Fakt/Beweis

Beweis

1. Es ist

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

2. Es ist wegen der Translationsinvarianz

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

3. Mit ${}s=c{\mathfrak {t}}$ ist

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

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