Properties of Fourier Transform/Scaling

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Theorem

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


Let $a$ be a non-zero real number.


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

$\map h x = \map f {a x}$


Then:

$\map {\hat h} s = \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}$


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


Proof

\(\ds \map {\hat h} s\) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x\)
\(\ds \) \(=\) \(\ds \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {a x} \rd x\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\size a} \int_{-\infty}^\infty e^{-2 \pi i \paren {a x} \paren {s / a} } \map f {a x} \map \rd {a x}\) Power of Power
\(\ds \) \(=\) \(\ds \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}\)

$\blacksquare$


Sources