 Solution

A cubic spline is a twice continuously differentiable function whose restriction to each knot interval is a polynomial of degree at most three. Its second derivative is continuous and piecewise linear. Natural endpoint conditions are $C^{\prime \prime }(x_0)=C^{\prime \prime }(x_n)=0$.

Linearity of both the piecewise-polynomial and smoothness conditions, and of the endpoint equations, makes the natural splines a vector space $V$. A continuous piecewise linear second derivative with zero endpoint values is determined freely by its $n-1$ interior nodal values. Integrating twice adds two arbitrary affine constants. HenceThis count includes the case of one interval.

An interpolating spline has $S(x_i)=f_i$ at every node. Set $e=T-S$, so $e$ vanishes at all nodes. On each open knot interval $S^{(3)}$ is constant. Two integrations by parts giveThe terms containing $e$ vanish at nodes. When summed, the interior $e^{\prime }{S}^{\prime \prime }$ terms cancel because $e^{\prime }$ and $S^{\prime \prime }$ are continuous. The two outer terms vanish by the natural endpoint conditions. ThusIntegrating the given square identity now yields the sharper energy decompositionIn particular the natural interpolating spline minimizes the squared-curvature integral.

If $T$ is another natural interpolant, the same minimizing argument in both directions makes the two energies equal. The final nonnegative integral then vanishes, so $T^{\prime \prime }=S^{\prime \prime }$ everywhere by continuity. Their difference is affine and zero at all knots, including the two distinct endpoints, so $T=S$. This proves uniqueness.

It also supplies existence if needed: nodal evaluation is a linear map from the $(n+1)$-dimensional space $V$ to ${\mathbb{R}}^{n+1}$. Its kernel is zero by the same argument with zero data and the zero competitor. It is therefore surjective, so every data vector has a natural spline interpolant.