Properties of Fourier Transform/Translation

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Theorem

Let $\map f x$ be a Lebesgue integrable function.


Let $x_0$ be a real number.


Let $\map h x$ be a Lebesgue integrable function such that:

$\map h x = \map f {x - x_0}$


Then:

$\map {\hat h} s = e^{-2 \pi i x_0 s} \map {\hat f} s$


where $\map {\hat h} s$ and $\map {\hat f} s$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.


Proof

\(\ds \map {\hat h} s\) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x\)
\(\ds \) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {x - x_0 } \rd x\)
\(\ds \) \(=\) \(\ds \int_{-\infty}^\infty \paren {e^{-2 \pi i x_0 s} } e^{-2 \pi i \paren {x - x_0 } s} \map f {x - x_0 } \rd x\)
\(\ds \) \(=\) \(\ds \paren {e^{-2 \pi i x_0 s} } \int_{-\infty}^\infty e^{-2 \pi i \paren {x - x_0 } s} \map f {x - x_0 } \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds e^{-2 \pi i x_0 s} \map {\hat f} s\)

$\blacksquare$


Sources