best approximation

One of the problems in approximation theory is to determine points that minimize distances (to a given point or subset). More precisely,

Problem - Let X be a metric space and SX a subset. Given x0X we want to know if there exists a point in S that minimizes the distance to x0, i.e. if there exists y0S such that


Definition - A point y0 that the above conditions is called a best approximation of x0 in S.

In general, best approximations do not exist. Thus, the problem is usually to identify classes of spaces X and S where the existence of best approximations can be assured.

Example : When S is compact, best approximations of a given point x0X in S always exist.

After one assures the existence of a best approximation, one can question about its uniqueness and how to calculate it explicitly.

Remark - There is no reason to restrict to metric spaces. The definition of best approximation can be given for pseudo-metric spaces, semimetric spaces or any other space where a suitable notion of ”distance” can be given.

Title best approximation
Canonical name BestApproximation
Date of creation 2013-03-22 17:31:23
Last modified on 2013-03-22 17:31:23
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Definition
Classification msc 41A50
Synonym optimal approximation