 Legendre polynomial kernel construction

For normalized Legendre polynomials, this kernel for density estimation reproduces evaluation at zero on every polynomial of degree below $\ell$: expand $p$ in the orthonormal basis to obtain $\int K_{\ell }p=p(0)$. Taking monomials proves integral one and zero moments of orders $1,\dots ,\ell -1$; compact support gives finite absolute moments. If exact order requires a nonzero moment of order $\ell$, add $({\varphi }_{\ell }(0)+1){\varphi }_{\ell }{1}_{[-1,1]}$. Lower moments remain unchanged. Writing $a_{\ell }=⟨u^{\ell },{\varphi }_{\ell }⟩\ne 0$ and evaluating the full expansion of $u^{\ell }$ at zero shows that the new moment is $a_{\ell }$. A useful three-derivative example is $K_3=(9-15u^2)1_{[-1,1]}/8$, whose first two moments vanish and whose squared L2 norm is $9/8$.