Triweight kernel ($K(x)=\frac{35}{32}(1-x^2{)}^3{1}_{|x|<1}$)

The triweight kernel is a compactly supported symmetric probability density with variance $1/9$. Its value and first two derivatives vanish at the support endpoints. Its integrated squared second derivative is $35$, and it minimizes that curvature among twice continuously differentiable densities with zero mean and the same variance, by the triweight density minimizes integrated squared curvature inequality.

Let $f_0$ be the triweight kernel and h a twice continuously differentiable density with matching zero mean and variance. Put $e=h-f_0$ and $c=35/32$. Two integrations by parts over its support give $\int e^{\prime \prime }{f}_0^{\prime \prime }=48c[h(1)+h(-1)]+c{\int }_{|x|>1}(360x^2-72)h(x)dx\ge 0$, using the zero mass and second moment of e. Therefore $R(h^{\prime \prime })=R(f_0^{\prime \prime })+R(e^{\prime \prime })+2\int e^{\prime \prime }{f}_0^{\prime \prime }\ge R(f_0^{\prime \prime })$. The endpoint terms matter because the third derivative jumps there.