Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 1 / c / Solution

For $x,y>0$, the Laplace transform identityexhibits $k_1$ as an inner product of the feature functions $t\mapsto e^{-tx}$ in $L^2(0,\infty )$. More explicitly, for any real $c_i$ and positive $x_i$,Therefore $k_1$ is a positive-semidefinite kernel.