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
- Weisstein, Eric W. "Fourier Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FourierTransform.html