Cesàro sums at factorial cutoffs track ordinary sums ($N_J=(J+1)!-1,{\sigma }_{N_J}-s_{N_J}=O(J^{-1})$)

For a formal cosine series ${\sum }_j{a}_j\cos (j!t)$ with uniformly bounded coefficients, the Fourier Cesàro mean at $N_J$ differs from its ordinary partial sum by $-(N_J+1{)}^{-1}{\sum }_{j\le J}j!a_j\cos (j!t)$. Since ${\sum }_{j\le J}j!\le 2J!$, 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.