Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 4 / a

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- Solution a

Solution

Assume first that

f

and

g

are integrable functions, so that Fubini's theorem justifies changing the order of the two integrals. In the defining Fourier transform of the convolution, put

y = x - x^{'}

:

h (k) = \int_{R} \int_{R} e^{- ik x} f (x - x^{'}) g (x^{'}) d x^{'} d x = \int_{R} \int_{R} e^{- ik (y + x^{'})} f (y) g (x^{'}) d y d x^{'} = (\int_{R} e^{- ik y} f (y) d y) (\int_{R} e^{- ik x^{'}} g (x^{'}) d x^{'}) .

This proves the convolution theorem with the stated normalization:

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

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