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

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- Solution a

Solution

For integrable functions

f, g

, Fubini's theorem justifies exchanging the two integrals in the Fourier transform of the convolution. Set

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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