Repeated Fourier Transform of Odd Function

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

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

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

Then:
 * $\map g t = -\map f t$