Cubic splines and variational characterization

Let $a=x_0<\cdots<x_m=b$ be distinct knots with $m\ge1$, and prescribe real values y at them. A cubic spline is polynomial of degree at most three on each interval and has two continuous derivatives across the interior knots. The natural interpolant also satisfies zero second derivative at both endpoints. These conditions determine a unique spline. Natural does not mean the endpoint slopes vanish, and it does not require the data themselves to arise from a function with zero endpoint curvature.

Denote the interval lengths by h and the nodal second derivatives by M. On an interval, the spline's second derivative is the line connecting the two endpoint second derivatives. Integrate twice and impose the two endpoint values to determine that cubic piece. Requiring neighboring first derivatives to agree then yields a tridiagonal linear system for the interior M values. Positive interval lengths make this natural system strictly diagonally dominant, so it is nonsingular; large variations in knot spacing still deserve numerical attention.

Compare the natural spline s with any real function u in $H^2([a,b])$ having the same nodal values. In one dimension these functions have representatives with continuous first derivatives, so nodal constraints and the following integration by parts are meaningful. Put v=u−s, which vanishes at every knot. On each interval, integrate the cross term involving s'' and v'' twice. The fourth derivative of each cubic piece vanishes, so only boundary terms remain.

Terms involving v vanish at knots because v is zero there. Interior terms involving v' cancel because s'' is continuous, and endpoint terms vanish by the natural zero-curvature conditions. Thus the cross term is zero, giving the energy identity below. Equality of energies forces v'' to vanish almost everywhere, so v is globally affine. Its values at the two distinct endpoints are zero, making it identically zero. The natural spline is therefore the unique minimizer, including the two-knot case.

The minimized quantity is an integrated squared second derivative, often called bending energy. It is a useful small-slope model of beam bending, not the exact geometric curvature energy of every parametric curve. The natural interpolant follows all data exactly, so a noisy measurement can still produce an undesirable feature. Minimal bending among exact interpolants does not imply that the original signal has been statistically recovered or that the spline preserves monotonicity and positivity.

A smoothing spline instead balances a data-misfit sum against a curvature penalty, with a chosen regularization parameter. That changes the optimization constraints and permits nonzero residuals at observations. Boundary conditions also matter: clamped splines prescribe endpoint slopes and minimize over the corresponding slope-constrained class, rather than the free-slope natural class. When choosing a method, identify whether the data are exact constraints or uncertain observations, and whether endpoint values, slopes, or periodicity carry actual modeling information.