Jackson operator estimate ($\Vert j_{n}f-f{\Vert }_{\infty }\le C{\omega }_2(f,1/n)$)

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 $E_N(f)\le C^{\prime }{N}^{-2}\Vert f^{\prime \prime }{\Vert }_{\infty }$.