Fourier Transform and its Inverse

Question

Could anyone show me how to prove the following results about Fourier Transform, please? It is stated in my book without proof. Thank you.

Let $\mathcal F$ denote the Fourier linear operator and $f$ be a $\mathcal L^1(\mathbb R)$ function. Then $$\mathcal F^2 (f) = f (-x).$$ That is, if we apply the Fourier transform twice, we get a spatially reversed version of the function.

Have a look here: http://math.stackexchange.com/questions/465565/proof-of-fourier-inverse-formula-for-l1-case . Also note that either you will have to interpret the "second" Fourier transform in the sense of distributions, or you will have to assume $\mathcal{F}f \in L^1$ as well. — PhoemueX, Jun 19 '14 at 08:32

score 3 · Accepted Answer · answered Jun 21 '14 at 11:57

3

If you use the following (unitary) definition of the Fourier transform

$$\mathcal{F}\{f(x)\}=F(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}dx\\ f(x)=\int_{-\infty}^{\infty}F(\xi)e^{2\pi ix\xi}d\xi$$

you have

$$\mathcal{F}^2\{f(x)\}=\mathcal{F}\{F(\xi)\}=\int_{-\infty}^{\infty}F(\xi)e^{-2\pi i x\xi}d\xi=f(-x)$$

answered Jun 21 '14 at 11:57

Matt L.

10,636

I don't see where you got the equation $f(x)=\int_{\infty}^{\infty} F( \xi ) e^{2 \pi i x \xi},d\xi$. – Arturo don Juan Jan 13 '15 at 05:51
@ArturoDonJuan: That's just the definition of the inverse Fourier transform. – Matt L. Jan 13 '15 at 07:52

Fourier Transform and its Inverse

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