 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/4/b/solution

Assume initially that

f, g

are integrable, so Fubini's theorem justifies interchanging the integrals. Transform the convolution and put

u = x - z

h (k) = \int_{R} \int_{R} f (z) g (x - z) e^{- ik x} d z d x = \int_{R} f (z) e^{- ik z} d z \int_{R} g (u) e^{- ik u} d u = f (k) g (k) .

This is the convolution theorem. No factor

2 π

appears in this direction because the forward transform in the question is unnormalized.

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