= Exact finite-grid high-frequency Jacobi smoothing optimum
{title2=$\omega_*=2/(a+b),\quad\mu_*=(b-a)/(a+b)$}
For the one-dimensional Dirichlet second-difference grid, high sine modes have eigenvalues $1-\omega t_k$, with $t_k=1-\cos(k\pi/(n+1))$ and $k\geq\lceil(n+1)/2\rceil$. Let $a$ and $b$ be the smallest and largest of these $t_k$. Convexity gives attenuation $\max(|1-\omega a|,|1-\omega b|)$. Balancing its endpoint values with opposite signs gives the displayed exact minimax choice. For odd $n$, $a=1$; for even $n$, $a=1+\sin(\pi/[2(n+1)])$, and always $b=1+\cos(\pi/(n+1))$. Only in the large-grid or uniform-in-grid limit do these become $\omega_*=2/3$, $\mu_*=1/3$.
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