Repeated Fourier Transform of Odd Function

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Theorem

Let $f: \R \to \R$ be an odd real function which is Lebesgue integrable.

Let $\ds \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$.

Let $\ds \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$.


Then:

$\map g t = -\map f t$


Proof

\(\ds \map g t\) \(=\) \(\ds \map f {-t}\) Repeated Fourier Transform of Real Function
\(\ds \) \(=\) \(\ds -\map f t\) Definition of Odd Function

$\blacksquare$


Sources