 Second derivative roughness penalty

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.