= Cesàro sums at factorial cutoffs track ordinary sums
{title2=$N_J=(J+1)!-1,\quad\sigma_{N_J}-s_{N_J}=O(J^{-1})$}
For a formal cosine <series> $\sum_ja_j\cos(j!t)$ with uniformly bounded coefficients, the Fourier <Cesàro mean> at $N_J$ differs from its ordinary partial sum by $-(N_J+1)^{-1}\sum_{j\leq J}j!a_j\cos(j!t)$. Since $\sum_{j\leq J}j!\leq2J!$, the difference is $O(1/J)$ uniformly in $t$. Thus any divergent behavior of these ordinary cutoff sums persists along the same Cesàro <subsequence>. Averaging by the sparse coefficient index is a different summation operation from averaging the actual Fourier frequency cutoffs.
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