Natural cubic smoothing spline

Assume AC. Let $[a,b]$ be a bounded interval, let $(x_i,y_i{)}_{i=1}^n\in ([a,b]\times \mathbb{R}{)}^n$, assume at least two distinct input values, and let $\lambda >0$. For $g\in W^{2,2}(a,b)$ use its unique continuous representative on $[a,b]$ supplied by the one-dimensional Sobolev representative theorem. The natural cubic smoothing spline is the unique minimizer, whose existence and uniqueness are proved in Existence, uniqueness, and spline reduction for cubic smoothing ↗, of $J(g)={\sum }_{i=1}^n(y_i-g(x_i){)}^2+\lambda {\int }_a^b\mathrm{\mid }{D}^{2}g(t){\mathrm{\mid }}^{2}dt\mathrm{.}$

The requirement of two distinct inputs is essential. The penalty is the squared $L^2$ norm of the second weak derivative. The data term includes every observation, including repeated inputs; only the spline knots are the distinct input values, as in Natural cubic splines.