Order of a kernel

In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.

Definitions
The literature knows two major definitions of the order of a kernel. Namely are:

Definition 1
Let $$ \ell \geq 1 $$ be an integer. Then, $$ K: \mathbb{R} \rightarrow \mathbb{R} $$ is a kernel of order $$ \ell $$ if the functions $$ u\mapsto u^{j}K(u), ~ j=0,1,...,\ell $$ are integrable and satisfy $$ \int K(u)du=1, ~ \int u^{j}K(u)du=0,~ ~j=1,...,\ell. $$