Splines replace one high-degree global polynomial by many low-degree pieces joined with controlled smoothness.
Spline space
Definition. A spline of degree \(r\) with knots \(a=t_0\lt t_1\lt\cdots\lt t_m=b\) is a function that is polynomial of degree at most \(r\) on each interval and satisfies prescribed continuity conditions at the knots.
Local support
Proposition. B-spline bases have local support: changing one coefficient affects only a few neighboring intervals. This locality gives sparse matrices, stable evaluation, and good behavior under local refinement.
Cubic splines
Theorem. The natural cubic spline interpolant minimizes \(\int_a^b |s''(x)|^2\,\mathrm{d}x\) among twice continuously differentiable functions interpolating the same data and satisfying the same endpoint conditions. Thus smoothness is encoded variationally.
Approximation order
Proposition. For sufficiently smooth \(f\), spline approximation error decays like a power of the maximal knot spacing. The rate is limited by both polynomial degree and smoothness of the target.
Why piecewise methods matter
Remark. Piecewise approximation handles functions with local features much better than global polynomial approximation. This is one reason splines sit naturally between interpolation, finite elements, and nonparametric smoothing.
Spline spaces and local support
Given knots \(a=t_0<\cdots<t_m=b\), a spline of degree \(p\) with simple interior knots is piecewise polynomial of degree at most \(p\) and belongs to \(C^{p-1}[a,b]\). B-splines \(B_{i,p}\) form a nonnegative partition-of-unity basis with compact support over \(p+2\) consecutive knots. Local support makes evaluation, assembly, and knot insertion sparse.
Cubic interpolation and variational characterization
Theorem (natural cubic spline). Given data \(y_i\) at knots \(t_i\), there is a unique \(C^2\) piecewise cubic interpolant \(s\) satisfying \[ s(t_i)=y_i,\qquad s''(a)=s''(b)=0. \] Among all \(f\in H^2(a,b)\) interpolating the data, it minimizes \[ \int_a^b|f''(x)|^2\,\mathrm{d}x. \]
Proof. Existence and uniqueness reduce to a strictly diagonally dominant tridiagonal system for the knot second derivatives. For the variational statement, write \(f=s+g\) with \(g(t_i)=0\). On each interval, \(s^{(4)}=0\). Twice integrating by parts and using continuity of \(s''\), interpolation zeros of \(g\), and natural boundary conditions gives \[ \int s''g''=0. \] Therefore \[ \int|f''|^2=\int|s''|^2+\int|g''|^2. \]
Error of cubic spline interpolation
For a quasi-uniform mesh of maximum width \(h\) and \(f\in C^4[a,b]\), the cubic spline interpolant satisfies \[ \|f-s\|_\infty\le Ch^4\|f^{(4)}\|_\infty, \] with derivative errors of order \(h^{4-k}\), \(k=1,2\), under compatible boundary conditions. Unlike high-degree global polynomial interpolation, refining the mesh preserves locality and avoids a rapidly growing Lebesgue constant.
Smoothing splines
For noisy observations \((x_i,y_i)\), the smoothing spline solves \[ \min_{f\in H^2} \sum_{i=1}^n\{y_i-f(x_i)\}^2 +\lambda\int|f''(x)|^2\,\mathrm{d}x. \] The representer theorem makes the minimizer a natural cubic spline with knots at distinct data locations. As \(\lambda\downarrow0\) it approaches interpolation; as \(\lambda\uparrow\infty\) it approaches a least-squares line. The effective degrees of freedom are \(\operatorname{tr}S_\lambda\) for the linear smoother matrix.
Adaptive piecewise approximation
Knots should concentrate where derivatives are large or regularity is low. Hierarchical B-splines and wavelets provide multiresolution refinement. Approximation order alone does not choose a mesh; a posteriori indicators or penalized model selection determine where extra degrees of freedom are useful.