Approximation theory

Approximation theory concerns the following problem: Given a function f(x) defined for x in a prescribed set X, a family of functions G, and a metric d(f, g) (a mathematical prescription for measuring the distance between two functions), determine a function g(x) in G that is "close" to f(x) for x in X. For computer applications, f(x) is typically a continuous function of one real variable, X is a real interval, G is a family of polynomials or of rational functions (ratios of polynomials), or of piecewise polynomials (splines), and the metric is either a least-squares metric