Jackson kernel ($J_n(t)$)

The normalized even kernelis nonnegative, has integral one, and is a trigonometric polynomial of degree $2(n-1)$. Its mass is concentrated on a scale $1/n$: $J_n(t)\le C\min (n,n^{-3}|t{|}^{-4})$ for $|t|\le \pi$. This controls a weighted second moment and produces the Jackson operator estimate.

For $j_{n}f(x)={\int }_{-\pi }^{\pi }f(x-t)J_n(t)dt$, 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 $\Vert j_{n}f-f{\Vert }_{\infty }\le C{\omega }_2(f,1/n)$. 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 $E_N(f)\le C^{\prime }{N}^{-2}\Vert f^{\prime \prime }{\Vert }_{\infty }$.