 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/4/a/solution

f, g

s = x - t

to obtain

h (k) = \int_{R} \int_{R} f (t) g (x - t) e^{- ik x} d t d x = \int_{R} \int_{R} f (t) g (s) e^{- ik (s + t)} d s d t = [\int_{R} f (t) e^{- ik t} d t] [\int_{R} g (s) e^{- ik s} d s] .

This proves the convolution theorem with the stated normalization:

h (k) = f (k) g (k) .

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