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''operator of best approximation''
 
''operator of best approximation''
  
A many-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636701.png" />, associating to each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636702.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636703.png" /> the set
+
A many-valued mapping $  P _ {M} : x \rightarrow P _ {M} x $,  
 +
associating to each element $  x $
 +
of a metric space $  X = ( X , \rho ) $
 +
the set
  
<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/m/m063/m063670/m0636704.png" /></td> </tr></table>
+
$$
 +
P _ {M} x  = \
 +
\{ {m \in M } : {\rho ( x , m ) = \rho ( x , M ) } \}
 +
$$
  
of elements of best approximation (cf. [[Element of best approximation|Element of best approximation]]) from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636705.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636706.png" /> is a [[Chebyshev set|Chebyshev set]], then the metric projection is a single-valued mapping. The problem of constructing an element of best approximation is often solved approximately, that is, an element is determined in the set
+
of elements of best approximation (cf. [[Element of best approximation|Element of best approximation]]) from the set $  M \subset  X $.  
 +
If $  M $
 +
is a [[Chebyshev set|Chebyshev set]], then the metric projection is a single-valued mapping. The problem of constructing an element of best approximation is often solved approximately, that is, an element is determined in the set
  
<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/m/m063/m063670/m0636707.png" /></td> </tr></table>
+
$$
 +
P _ {M}  ^ {t} x  = \
 +
\{ {m \in M } : {\rho ( x , m ) \leq  t + \rho ( x , M ) } \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636708.png" /> is sufficiently small. From the properties of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m0636709.png" /> it is sometimes possible to obtain properties of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367010.png" />. E.g., if for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367011.png" /> of a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367012.png" /> a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367013.png" /> exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367014.png" /> is convex (connected), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367015.png" /> is convex (respectively, connected).
+
where $  t > 0 $
 +
is sufficiently small. From the properties of the mapping $  P _ {M}  ^ {t} : x \rightarrow P _ {M}  ^ {t} x $
 +
it is sometimes possible to obtain properties of the set $  M $.  
 +
E.g., if for any element $  x $
 +
of a normed space $  X $
 +
a number $  t = t ( x) > 0 $
 +
exists such that $  P _ {M}  ^ {t} x $
 +
is convex (connected), then $  M $
 +
is convex (respectively, connected).
  
From the point of view of applications it is useful to know whether the metric projection has such properties as linearity, continuity, uniform continuity, etc. A metric projection on a Chebyshev subspace of a normed space is, in general, not linear. If the metric projection on each subspace of fixed dimension is single-valued and linear, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367016.png" /> is linearly isometric to an inner-product space. The metric projection on a non-empty [[Approximately-compact set|approximately-compact set]] in a metric space is upper semi-continuous; in particular, in a normed space the metric projection onto a finite-dimensional Chebyshev subspace is continuous; the metric projection may be not lower semi-continuous if the subspace is not Chebyshev. There exists a reflexive strictly-convex space and an infinite-dimensional subspace on which the metric projection is discontinuous. The metric projection on any closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367017.png" /> in a Hilbert space satisfies a Lipschitz condition:
+
From the point of view of applications it is useful to know whether the metric projection has such properties as linearity, continuity, uniform continuity, etc. A metric projection on a Chebyshev subspace of a normed space is, in general, not linear. If the metric projection on each subspace of fixed dimension is single-valued and linear, then $  X $
 +
is linearly isometric to an inner-product space. The metric projection on a non-empty [[Approximately-compact set|approximately-compact set]] in a metric space is upper semi-continuous; in particular, in a normed space the metric projection onto a finite-dimensional Chebyshev subspace is continuous; the metric projection may be not lower semi-continuous if the subspace is not Chebyshev. There exists a reflexive strictly-convex space and an infinite-dimensional subspace on which the metric projection is discontinuous. The metric projection on any closed convex set $  M $
 +
in a Hilbert space satisfies a Lipschitz condition:
  
<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/m/m063/m063670/m06367018.png" /></td> </tr></table>
+
$$
 +
\| P _ {M} x - P _ {M} y \|  \leq  K  \| x - y \| ,
 +
$$
  
with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063670/m06367019.png" />.
+
with constant $  K = 1 $.
  
 
The continuity property of a metric projection and its generalizations have found applications in ill-posed problems, in the convexity problem for Chebyshev sets, in the construction of elements of best approximation, etc.
 
The continuity property of a metric projection and its generalizations have found applications in ill-posed problems, in the convexity problem for Chebyshev sets, in the construction of elements of best approximation, etc.

Latest revision as of 08:00, 6 June 2020


operator of best approximation

A many-valued mapping $ P _ {M} : x \rightarrow P _ {M} x $, associating to each element $ x $ of a metric space $ X = ( X , \rho ) $ the set

$$ P _ {M} x = \ \{ {m \in M } : {\rho ( x , m ) = \rho ( x , M ) } \} $$

of elements of best approximation (cf. Element of best approximation) from the set $ M \subset X $. If $ M $ is a Chebyshev set, then the metric projection is a single-valued mapping. The problem of constructing an element of best approximation is often solved approximately, that is, an element is determined in the set

$$ P _ {M} ^ {t} x = \ \{ {m \in M } : {\rho ( x , m ) \leq t + \rho ( x , M ) } \} , $$

where $ t > 0 $ is sufficiently small. From the properties of the mapping $ P _ {M} ^ {t} : x \rightarrow P _ {M} ^ {t} x $ it is sometimes possible to obtain properties of the set $ M $. E.g., if for any element $ x $ of a normed space $ X $ a number $ t = t ( x) > 0 $ exists such that $ P _ {M} ^ {t} x $ is convex (connected), then $ M $ is convex (respectively, connected).

From the point of view of applications it is useful to know whether the metric projection has such properties as linearity, continuity, uniform continuity, etc. A metric projection on a Chebyshev subspace of a normed space is, in general, not linear. If the metric projection on each subspace of fixed dimension is single-valued and linear, then $ X $ is linearly isometric to an inner-product space. The metric projection on a non-empty approximately-compact set in a metric space is upper semi-continuous; in particular, in a normed space the metric projection onto a finite-dimensional Chebyshev subspace is continuous; the metric projection may be not lower semi-continuous if the subspace is not Chebyshev. There exists a reflexive strictly-convex space and an infinite-dimensional subspace on which the metric projection is discontinuous. The metric projection on any closed convex set $ M $ in a Hilbert space satisfies a Lipschitz condition:

$$ \| P _ {M} x - P _ {M} y \| \leq K \| x - y \| , $$

with constant $ K = 1 $.

The continuity property of a metric projection and its generalizations have found applications in ill-posed problems, in the convexity problem for Chebyshev sets, in the construction of elements of best approximation, etc.

References

[1] I.M. Singer, "The theory of best approximation and functional analysis" , CBMS Regional Conf. Ser. , 13 , SIAM (1974)
[2] L.P. Vlasov, "Approximative properties of sets in normed linear spaces" Russian Math. Surveys , 28 : 6 (1973) pp. 3–66 Uspekhi Mat. Nauk , 28 : 6 (1973) pp. 3–66
[3] V.I. Berdyshev, "Uniform continuity of the metric projection and of the -projection" , Theory of Approximation of Functions. Internat. Conf., Kaluga 1975 , Moscow (1977) pp. 37–41 (In Russian)
How to Cite This Entry:
Metric projection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric_projection&oldid=13496
This article was adapted from an original article by V.I. Berdyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article