BestApproximation 2007-09-02 18:08:56 2007-09-02 18:08:56 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here One of the \PMlinkescapetext{central} problems in approximation theory is to determine points that minimize distances (to a given point or subset). More precisely, {\bf 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 \begin{displaymath} d(x_0,y_0)=\inf_{y \in S}d(x_0,y) \end{displaymath} {\bf Definition -} A point $y_0$ that \PMlinkescapetext{satisfies} the above conditions is called a {\bf 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. {\bf 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. {\bf 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.