= Minimum roughness property of a clamped cubic spline
{title2=$J(T)=J(S)+J(T-S),\qquad J(u)=\int_{x_0}^{x_n}(u'')^2\,dx$}
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 $T'$ 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 $\int S''E''=[S''E'-S'''E]$, since the cubic piece has zero <fourth derivative>. Terms containing $E$ vanish at the <spline knots>; terms containing $E'$ cancel at interior <spline knots> because $S''$ and $E'$ are continuous. The endpoint terms vanish because $E'$ is zero there. Thus $\int S''E''=0$, and expanding $(S''+E'')^2$ proves the identity. Equality makes $E''=0$, 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 $E'$ 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 $\int ET^{(4)}=\int(E'')^2\geq0$.
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