 Solution

ID: 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/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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