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Difference between revisions of "Chebyshev point"

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''of a system of linear inequalities
 
''of a system of linear inequalities
  
<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/c/c021/c021930/c0219301.png" /></td> </tr></table>
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$$\eta_i(x)=a_{i1}\xi_1+\dots+a_{in}\xi_n+a_i\leq0,\quad i=1,\dots,m,$$
  
 
''
 
''
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021930/c0219302.png" /> at which the minimax
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A point $x=(\xi_1,\dots,xi_n)$ at which the minimax
  
<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/c/c021/c021930/c0219303.png" /></td> </tr></table>
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$$\min_x\max_{i\leq i\leq m}\eta_i(x)$$
  
 
is attained. The problem of finding a Chebyshev point reduces to the general problem of linear programming [[#References|[1]]].
 
is attained. The problem of finding a Chebyshev point reduces to the general problem of linear programming [[#References|[1]]].
  
A more general notion is that of a Chebyshev point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021930/c0219304.png" /> of a system of hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021930/c0219305.png" /> in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021930/c0219306.png" />, i.e. a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021930/c0219307.png" /> for which
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A more general notion is that of a Chebyshev point $x^*$ of a system of hyperplanes $\{H_i\}_{i-1}^m$ in a Banach space $X$, i.e. a point $x^*$ for which
  
<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/c/c021/c021930/c0219308.png" /></td> </tr></table>
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$$\sup_{1\leq i\leq m}\inf_{z\in H_i}\|z-x^*\|=\inf_{x\in X}\sup_{1\leq i\leq m}\inf_{z\in H_i}\|z-x\|.$$
  
 
Chebyshev points are often chosen as  "solutions"  of incompatible linear systems of equations and inequalities.
 
Chebyshev points are often chosen as  "solutions"  of incompatible linear systems of equations and inequalities.
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====Comments====
 
====Comments====
The term  "Chebyshev point"  or  "Chebyshev nodeChebyshev node"  is also used to denote a zero of a Chebyshev polynomial (cf. [[Chebyshev polynomials|Chebyshev polynomials]]) in the theory of (numerical) interpolation, integration, etc. [[#References|[a1]]].
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The term  "Chebyshev point"  or  "Chebyshev node"  is also used to denote a zero of a Chebyshev polynomial (cf. [[Chebyshev polynomials|Chebyshev polynomials]]) in the theory of (numerical) interpolation, integration, etc. [[#References|[a1]]].
  
 
Sometimes Chebyshev is spelled differently as Tschebyshev or Tschebycheff.
 
Sometimes Chebyshev is spelled differently as Tschebyshev or Tschebycheff.

Revision as of 19:43, 21 November 2018

of a system of linear inequalities

$$\eta_i(x)=a_{i1}\xi_1+\dots+a_{in}\xi_n+a_i\leq0,\quad i=1,\dots,m,$$

A point $x=(\xi_1,\dots,xi_n)$ at which the minimax

$$\min_x\max_{i\leq i\leq m}\eta_i(x)$$

is attained. The problem of finding a Chebyshev point reduces to the general problem of linear programming [1].

A more general notion is that of a Chebyshev point $x^*$ of a system of hyperplanes $\{H_i\}_{i-1}^m$ in a Banach space $X$, i.e. a point $x^*$ for which

$$\sup_{1\leq i\leq m}\inf_{z\in H_i}\|z-x^*\|=\inf_{x\in X}\sup_{1\leq i\leq m}\inf_{z\in H_i}\|z-x\|.$$

Chebyshev points are often chosen as "solutions" of incompatible linear systems of equations and inequalities.

References

[1] S.I. [S.I. Zukhovitskii] Zukhovitsky, L.I. Avdeeva, "Linear and convex programming" , Saunders (1966)
[2] P.K. Belobrov, "The Chebyshev point of a system of translates of subspaces in a Banach space" Mat. Zametki , 8 : 4 (1970) pp. 29–40 (In Russian)
[3] I.I. Eremin, "Incompatible systems of linear inequalities" Dokl. Akad. Nauk SSSR , 138 : 6 (1961) pp. 1280–1283 (In Russian)


Comments

The term "Chebyshev point" or "Chebyshev node" is also used to denote a zero of a Chebyshev polynomial (cf. Chebyshev polynomials) in the theory of (numerical) interpolation, integration, etc. [a1].

Sometimes Chebyshev is spelled differently as Tschebyshev or Tschebycheff.

References

[a1] L. Fox, I. Parker, "Chebyshev polynomials in numerical analysis" , Oxford Univ. Press (1968)
How to Cite This Entry:
Chebyshev point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_point&oldid=14353
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article