= Jackson operator estimate
{c}
{title2=$\|j_nf-f\|_\infty\leq C\omega_2(f,1/n)$}
For $j_nf(x)=\int_{-\pi}^{\pi}f(x-t)J_n(t)\,dt$, evenness and unit mass reduce the error to an integral of second differences. The <Jackson kernel> bound and the scaling inequality for the <second modulus of smoothness> yield $\|j_nf-f\|_\infty\leq C\omega_2(f,1/n)$. Since the degree is $2(n-1)$, a degree-$N$ estimate uses $n=\lfloor N/2\rfloor+1$. For $f\in C^2$ this gives $E_N(f)\leq C'N^{-2}\|f''\|_\infty$.
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