Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 39B / iv / Solution

Let $k_0=\lceil (n+1)/2\rceil$, $\theta =\pi /(n+1)$, and define the high-frequency endpoint parametersThey obey $1\le a\le b<2$ (with equality of endpoints for a single high mode). Since $|1-\omega t|$ is convex in $t$ and the endpoint modes are included, the exact high-frequency attenuation isThe minimax choice balances the endpoint values with opposite signs. If they were not balanced, moving $\omega$ toward their balancing point would lower the larger endpoint value. ThusFor odd $n$, $a=1$, giving ${\omega }_{\ast }=2/(2+\cos \theta )$ and ${\mu }_{\ast }=\cos \theta /(2+\cos \theta )$. For even $n$, $a=1+\sin (\theta /2)$, givingThese give the exact finite-grid high-frequency Jacobi smoothing optimum, including complete annihilation when there is only one high-frequency mode. Their grid-independent limits are $\boxed{{\omega }_{\ast }=2/3,{\mu }_{\ast }=1/3}$. Choosing $2/3$ damps every high mode by at most $1/3$ on every grid, while its low-mode factor remains $1-O(n^{-2})$. Thus high frequencies decay geometrically much faster. The familiar $2/3,1/3$ pair is the optimal uniform-in-grid smoother; it need not be the exact optimum on a particular finite sine-mode grid.