 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/4/c/solution

(g * g) (x)

is the length of the overlap of

(- 1, 1)

with

(x - 1, x + 1)

. This overlap has length

2 - ∣ x ∣

when

∣ x ∣ < 2

and zero otherwise, so

u = g * g

u (k) = g (k)^{2} = \frac{4 sin ^{2} k}{k ^{2}}, u (0) = 4.

The value at zero is also the area of the triangular function, providing a normalization check on its Fourier transform.

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