Unit-width box kernel ($K(u)=1_{[-1/2,1/2]}(u)$)

The unit-width centred box kernel is $K(u)=1_{[-1/2,1/2]}(u)$. It has integral one and squared $L^2$ norm one. Its rescaling $K_h(u)=h^{-1}K(u/h)$ averages a function on intervals of length $h$. It is obtained from the uniform smoothing kernel with support $[-1,1]$ by rescaling, and is useful in both kernel density estimation and local polynomial regression.