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For $\lambda >0$, minimize the smoothing spline objective over the Sobolev space of functions with square-integrable second derivative on the observed interval:The roughness penalty measures curvature; its null space consists of affine functions. To see why the minimizer is a natural cubic spline, let $g$ be the natural spline interpolating the values of any competitor $m$ at its distinct observation locations, and put $h=m-g$. Integration by parts twice on each knot interval gives $\int g^{\prime \prime }{h}^{\prime \prime }=0$: $g^{\prime \prime \prime \prime }=0$ on each interval, $h$ vanishes at the knots, $g^{\prime \prime }$ is continuous, and $g^{\prime \prime }$ vanishes at the outer endpoints. Hencewhile both functions have the same residual sum of squares. This proves the reduction to a finite-dimensional spline space.

For distinct $X_i$ and $n\ge 2$, use its $n$-dimensional basis, write $B_{ij}=b_j(X_i)$, and define the positive semidefinite matrixThe penalized least squares problem is $\Vert Y-B\beta {\Vert }^2+\lambda {\beta }^T\Omega \beta$, soUse linear continuation outside the observation interval. The fitted-value matrix is $S_{\lambda }=B(B^{T}B+\lambda \Omega {)}^{-1}{B}^T$; its trace gives the effective degrees of freedom. As $\lambda$ decreases to zero, the natural spline approaches interpolation, with trace $n$; as it grows without bound, the fit approaches ordinary least squares on $1,x$, with trace two. For $\lambda =0$ the criterion alone does not determine values between observations; the natural interpolant is the limiting convention. Repeated observation locations are combined by using distinct knots and their multiplicities as weights, rather than pretending the spline design has $n$ independent interpolation values.