Cubic smoothing spline (${\hat{g}}_{\lambda }=N(I+\lambda \Gamma {)}^{-1}\mathbf{Y}$)

For $\lambda >0$ and at least two distinct design points, the unique $C^2[a,b]$ minimizer of squared residuals plus $\lambda J(g)$ is $N(I+\lambda \Gamma {)}^{-1}\mathbf{Y}$. The minimum roughness property of the natural cubic spline interpolant reduces the optimization to the value vector. The matrix $I+\lambda \Gamma$ is a positive-definite matrix, so the normal equations have a unique solution; the interpolation equality case gives uniqueness as a function. Linear tails are part of the natural boundary convention.