structural:reformatting, structural:summarization
Function approximation

Interpolation is a common way to approximate functions. Given a function f: → R {\displaystyle f:\to \mathbb {R} } with a set of points x 1, x 2, …, x n ∈ {\displaystyle x_{1},x_{2},\dots,x_{n}\in } one can form a function s: → R {\displaystyle s:\to \mathbb {R} } such that f (x i) = s (x i) {\displaystyle f(x_{i})=s(x_{i})} for i = 1, 2, …, n {\displaystyle i=1,2,\dots,n} (that is, that s {\displaystyle s} interpolates f {\displaystyle f} 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 f ∈ C 4 () {\displaystyle f\in C^{4}()} (four times continuously differentiable) then cubic spline interpolation has an error bound given by ‖ f − s ‖ ∞ ≤ C ‖ f (4) ‖ ∞ h 4 {\displaystyle \|f-s\|_{\infty }\leq C\|f^{(4)}\|_{\infty }h^{4}} where h max i = 1, 2, …, n − 1 | x i + 1 − x i | {\displaystyle h\max _{i=1,2,\dots,n-1}|x_{i+1}-x_{i}|} and C {\displaystyle C} is a constant.