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Year 2 of the natural sciences course of the University of Cambridge

Mathematics course of the year 2 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge

2025

Paper 1

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 $y=x-x'$:
$$
\begin{aligned}
\widetilde h(k)
&=\int_{\mathbb R}\int_{\mathbb R}e^{-ikx}f(x-x')g(x')\,dx'\,dx\\
&=\int_{\mathbb R}\int_{\mathbb R}e^{-ik(y+x')}f(y)g(x')\,dy\,dx'\\
&=\left(\int_{\mathbb R}e^{-iky}f(y)\,dy\right)\left(\int_{\mathbb R}e^{-ikx'}g(x')\,dx'\right).
\end{aligned}
$$
This proves the <convolution theorem> with the stated normalization:
$$
\boxed{\widetilde h(k)=\widetilde f(k)\widetilde g(k).}
$$

Solution

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 / Solution

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 / Solution

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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 / Solution

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