# 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 $S\subseteq X$ a subset. Given ${x}_{0}\in X$ we want to know if there exists a point in $S$ that minimizes the distance to ${x}_{0}$, i.e. if there exists ${y}_{0}\in S$ such that

$$d({x}_{0},{y}_{0})=\underset{y\in S}{inf}d({x}_{0},y)$$ |

Definition - A point ${y}_{0}$ that the above conditions is called a best approximation of ${x}_{0}$ 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 ${x}_{0}\in X$ 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 |