Jackson kernel
= Jackson kernel
{c}
{title2=$J_n(t)$}
The normalized even kernel
$$
J_n(t)=\frac{3}{2\pi n(2n^2+1)}\left(\frac{\sin(nt/2)}{\sin(t/2)}\right)^4
$$
is nonnegative, has integral one, and is a <trigonometric polynomial> of degree $2(n-1)$. Its mass is concentrated on a scale $1/n$: $J_n(t)\leq C\min(n,n^{-3}|t|^{-4})$ for $|t|\leq\pi$. This controls a weighted second moment and produces the <Jackson operator estimate>.