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Difference between revisions of "Chebyshev alternation"

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A property of the difference between a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218201.png" /> on a closed set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218202.png" /> and a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218203.png" /> (in a [[Chebyshev system|Chebyshev system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218204.png" />) on an ordered sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218205.png" /> points
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A property of the difference between a continuous function $f(x)$ on a closed set of real numbers $Q$ and a polynomial $P_n(x)$ (in a [[Chebyshev system|Chebyshev system]] $\{\phi_k(x)\}_0^n$) on an ordered sequence of $n+2$ points
  
<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/c021820/c0218206.png" /></td> </tr></table>
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$$\{x_i\}_0^{n+1}\subset Q,\quad x_0<\dotsb<x_{n+1},$$
  
 
such that
 
such that
  
<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/c021820/c0218207.png" /></td> </tr></table>
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$$f(x_i)-P_n(x_i)=(-1)^i\epsilon\|f(x)-P_n(x)\|_{C(Q)},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218208.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c0218209.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021820/c02182010.png" /> are called Chebyshev alternation points or points in Chebyshev alternation (cf. also [[Alternation, points of|Alternation, points of]]).
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where $\epsilon=1$ or $-1$. The points $\{x_i\}_0^{n+1}$ are called Chebyshev alternation points or points in Chebyshev alternation (cf. also [[Alternation, points of|Alternation, points of]]).
  
  

Latest revision as of 12:37, 14 February 2020

A property of the difference between a continuous function $f(x)$ on a closed set of real numbers $Q$ and a polynomial $P_n(x)$ (in a Chebyshev system $\{\phi_k(x)\}_0^n$) on an ordered sequence of $n+2$ points

$$\{x_i\}_0^{n+1}\subset Q,\quad x_0<\dotsb<x_{n+1},$$

such that

$$f(x_i)-P_n(x_i)=(-1)^i\epsilon\|f(x)-P_n(x)\|_{C(Q)},$$

where $\epsilon=1$ or $-1$. The points $\{x_i\}_0^{n+1}$ are called Chebyshev alternation points or points in Chebyshev alternation (cf. also Alternation, points of).


Comments

Points in Chebyshev alternation are also called Chebyshev points of alternation, and their set is also called an alternating set. See also (the references in) Alternation, points of.

References

[a1] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2, Sect. 6
How to Cite This Entry:
Chebyshev alternation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_alternation&oldid=13923
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article