= Minimum roughness property with absolutely continuous first derivatives
{title2=$\int(f'')^2=\int(g'')^2+\int(f''-g'')^2$}
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 $\int g''h''=0$: 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.
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