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Difference between revisions of "Element of best approximation"

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An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352601.png" /> in a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352602.png" /> that is a [[Best approximation|best approximation]] to a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352603.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352604.png" />, i.e. is such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352605.png" /></td> </tr></table>
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An element $u_0$ in a given set $F$ that is a [[best approximation]] to a given element $x$ in a [[metric space]] $X$, i.e. is such that
 
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$$
This is a generalization of the classical concept of a [[Polynomial of best approximation|polynomial of best approximation]]. The main questions concerning elements of best approximation are: their existence and uniqueness, their characteristic properties (see [[Chebyshev theorem|Chebyshev theorem]]), the properties of the operator that associates with each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352606.png" /> the set of elements of best approximation (see [[Metric projection|Metric projection]]; [[Approximately-compact set|Approximately-compact set]]), and numerical methods for the construction of elements of best approximation.
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\rho(u_0,x) = \inf \{ \rho(u,x) : x \in F \} \ .
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$$
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This is a generalization of the classical concept of a [[polynomial of best approximation]]. The main questions concerning elements of best approximation are: their existence and uniqueness, their characteristic properties (see [[Chebyshev theorem]]), the properties of the operator that associates with each element $x \in X$ the set of elements of best approximation (see [[Metric projection]]; [[Approximately-compact set]]), and numerical methods for the construction of elements of best approximation.

Latest revision as of 20:09, 27 December 2014

2020 Mathematics Subject Classification: Primary: 41A50 [MSN][ZBL]

An element $u_0$ in a given set $F$ that is a best approximation to a given element $x$ in a metric space $X$, i.e. is such that $$ \rho(u_0,x) = \inf \{ \rho(u,x) : x \in F \} \ . $$ This is a generalization of the classical concept of a polynomial of best approximation. The main questions concerning elements of best approximation are: their existence and uniqueness, their characteristic properties (see Chebyshev theorem), the properties of the operator that associates with each element $x \in X$ the set of elements of best approximation (see Metric projection; Approximately-compact set), and numerical methods for the construction of elements of best approximation.

How to Cite This Entry:
Element of best approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Element_of_best_approximation&oldid=18961
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article