The kernel is the <Brownian bridge covariance kernel>. If $C_X\phi=\lambda\phi$, differentiating the integral equation twice gives
$$
\lambda\phi''(t)=-\phi(t),\qquad \phi(0)=\phi(1)=0.
$$
The normalized solutions and eigenvalues are therefore
$$
\boxed{\phi_k(t)=\sqrt2\sin(k\pi t),\qquad
\lambda_k=\frac1{k^2\pi^2}},\qquad k\geq1.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-225/1/i/solution