Editing Interpolation (section)

==Function approximation==
Interpolation is a common way to approximate functions. Given a function <math>f:[a,b] \to \mathbb{R}</math> with a set of points <math>x_1, x_2, \dots, x_n \in [a, b]</math> one can form a function <math>s: [a,b] \to \mathbb{R}</math> such that <math>f(x_i)=s(x_i)</math> for <math>i=1, 2, \dots, n</math> (that is, that <math>s</math> interpolates <math>f</math> at these points). In general, an interpolant need not be a good approximation, but there are well known and often reasonable conditions where it will. For example, if <math>f\in C^4([a,b])</math> (four times continuously differentiable) then [[spline interpolation|cubic spline interpolation]] has an error bound given by <math>\|f-s\|_\infty \leq C \|f^{(4)}\|_\infty h^4</math> where <math>h \max_{i=1,2, \dots, n-1} |x_{i+1}-x_i|</math> and <math>C</math> is a constant.<ref>{{cite journal |last1=Hall |first1=Charles A. |last2=Meyer |first2=Weston W. |title=Optimal Error Bounds for Cubic Spline Interpolation |journal=Journal of Approximation Theory |date=1976 |volume=16 |issue=2 |pages=105–122 |doi=10.1016/0021-9045(76)90040-X |doi-access=free }}</ref>