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
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem