Minimum roughness property with absolutely continuous first derivatives ($\int (f^{\prime \prime }{)}^2=\int (g^{\prime \prime }{)}^2+\int (f^{\prime \prime }-g^{\prime \prime }{)}^2$)

The minimum roughness property of the natural cubic spline interpolant extends to interpolants with an absolutely continuous first derivative and square-integrable second derivative. For $h=f-g$ vanishing at the knots, integration by parts on each cubic interval gives $\int g^{\prime \prime }{h}^{\prime \prime }=0$: the knot boundary terms cancel by continuity, the exterior terms vanish by natural boundary conditions, and the remaining constant-third-derivative term vanishes because $h$ is zero at the interval endpoints. Expanding the square proves minimality. Equality makes $h$ affine; its zeros at two distinct knots force $h=0$, proving uniqueness. Competitors with infinite roughness cannot improve the finite spline penalty.