Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 2 / c / Solution

Extend $f$ by zero outside $[0,1]$. Since the Fourier transform of $e^{-|x|}$ is $2/(1+{\xi }^2)>0$,for $f\ne 0$. Thus every eigenvalue is positive. From the relation in part (b),The transcendental equation has its successive roots in intervals separated by the poles and zeros of $\tan \omega$, so ${\omega }_n$ grows linearly with $n$. Hence