 Minimum roughness property of a clamped cubic spline

Let $S$ be a clamped cubic spline interpolant and let $T$ have the same values at all spline knots and the same endpoint slopes. For $T\in C^2[x_0,x_n]$, or more generally $T^{\prime }$ an absolutely continuous function with square-integrable second derivative, the exact displayed identity holds. In particular $S$ uniquely minimizes the second derivative roughness penalty $J$ among these competitors.

To prove the identity, put $E=T-S$. On each knot interval, two applications of integration by parts give $\int S^{\prime \prime }{E}^{\prime \prime }=[S^{\prime \prime }{E}^{\prime }-S^{\prime \prime \prime }E]$, since the cubic piece has zero fourth derivative. Terms containing $E$ vanish at the spline knots; terms containing $E^{\prime }$ cancel at interior spline knots because $S^{\prime \prime }$ and $E^{\prime }$ are continuous. The endpoint terms vanish because $E^{\prime }$ is zero there. Thus $\int S^{\prime \prime }{E}^{\prime \prime }=0$, and expanding $(S^{\prime \prime }+E^{\prime \prime }{)}^2$ proves the identity. Equality makes $E^{\prime \prime }=0$, so $E$ is an affine function; a prescribed endpoint value and slope force $E=0$.

If $T$ also matches the first derivative of $S$ at every spline knot, both $E$ and $E^{\prime }$ vanish at the ends of each knot interval. For a sufficiently differentiable $T$, integration by parts separately on that interval gives the stronger local identity $\int E{T}^{(4)}=\int (E^{\prime \prime }{)}^2\ge 0$.