Second derivative roughness penalty
= Second derivative roughness penalty
{title2=$J(g)=\int_a^b(g^{\prime\prime})^2\,dx$}
The nonnegative functional $J(g)=\int_a^b(g^{\prime\prime}(x))^2\,dx$. Its <null space> on $C^2[a,b]$ consists of <affine functions>. The <natural cubic spline interpolant> uniquely minimizes this penalty among functions matching at least two prescribed knot values. It is a squared <second derivative> penalty, not the exact geometric curvature energy of a graph.