Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 4 / a

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Solution

Using Fourier inversion with the stated convention,

f (- ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (t) e^{- iω t} d t .

Hence applying the transform twice gives the duality property of the Fourier transform:

F [f] (ω) = 2 π f (- ω) .

The formula holds directly for rapidly decreasing smooth functions, and extends through the usual L2 space or tempered distribution formulations when absolute convergence is unavailable.

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