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Difference between revisions of "Chebyshev theorem"

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If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c0219901.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c0219902.png" /> and if
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{{TEX|done}}
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{{MSC|41A50}}
  
<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/c021990/c0219903.png" /></td> </tr></table>
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If a function $f(x)$ is continuous on $[a,b]$ and if
  
<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/c021990/c0219904.png" /></td> </tr></table>
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$$A=\max_{a\leq x\leq b}|f(x)-P_n(x)|,$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c0219905.png" /> is the polynomial of best uniform approximation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c0219906.png" />, i.e.
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$$P_n(x)=\sum_{k=0}^na_kx^k,$$
  
<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/c021990/c0219907.png" /></td> </tr></table>
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then $P_n(x)$ is the polynomial of best uniform approximation for $f(x)$, i.e.
  
if and only if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c0219908.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c0219909.png" /> in [[Chebyshev alternation|Chebyshev alternation]], which means that the condition
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$$\max_{a\leq x\leq b}|f(x)-P_n(x)|=\min_{\{c_k\}}\max_{a\leq x\leq b}\left|f(x)-\sum_{k=0}^nc_kx^k\right|,$$
  
<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/c021990/c02199010.png" /></td> </tr></table>
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if and only if there exist $n+2$ points $a\leq x_0<\dots<x_{n+1}\leq b$ in [[Chebyshev alternation|Chebyshev alternation]], which means that the condition
  
is satisfied, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c02199011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c02199012.png" />. This theorem was proved by P.L. Chebyshev in 1854 (cf. [[#References|[1]]]) in a more general form, namely for the best uniform approximation of functions by rational functions with fixed degrees of the numerator and denominator. Chebyshev's theorem remains valid if instead of algebraic polynomials one considers polynomials
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$$f(x_i)-P_n(x_i)=\epsilon A(-1)^i,\quad i=0,\dots,n+1,$$
  
<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/c021990/c02199013.png" /></td> </tr></table>
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is satisfied, where $\epsilon=1$ or $-1$. This theorem was proved by P.L. Chebyshev in 1854 (cf. [[#References|[1]]]) in a more general form, namely for the best uniform approximation of functions by rational functions with fixed degrees of the numerator and denominator. Chebyshev's theorem remains valid if instead of algebraic polynomials one considers polynomials
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021990/c02199014.png" /> is a [[Chebyshev system|Chebyshev system]]. The criterion formulated in Chebyshev's theorem leads to methods for the approximate construction of polynomials of best uniform (Chebyshev) approximation. In a somewhat different formulation Chebyshev's theorem can be extended to functions of a complex variable (cf. [[#References|[2]]]) and to abstract functions (cf. [[#References|[3]]]).
+
$$P_n(x)=\sum_{k=0}^nc_k\phi_k(x),$$
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 +
where $\{\phi_k(x)\}_{k=0}^n$ is a [[Chebyshev system|Chebyshev system]]. The criterion formulated in Chebyshev's theorem leads to methods for the approximate construction of polynomials of best uniform (Chebyshev) approximation. In a somewhat different formulation Chebyshev's theorem can be extended to functions of a complex variable (cf. [[#References|[2]]]) and to abstract functions (cf. [[#References|[3]]]).
  
 
====References====
 
====References====

Latest revision as of 17:05, 28 June 2015

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

If a function $f(x)$ is continuous on $[a,b]$ and if

$$A=\max_{a\leq x\leq b}|f(x)-P_n(x)|,$$

$$P_n(x)=\sum_{k=0}^na_kx^k,$$

then $P_n(x)$ is the polynomial of best uniform approximation for $f(x)$, i.e.

$$\max_{a\leq x\leq b}|f(x)-P_n(x)|=\min_{\{c_k\}}\max_{a\leq x\leq b}\left|f(x)-\sum_{k=0}^nc_kx^k\right|,$$

if and only if there exist $n+2$ points $a\leq x_0<\dots<x_{n+1}\leq b$ in Chebyshev alternation, which means that the condition

$$f(x_i)-P_n(x_i)=\epsilon A(-1)^i,\quad i=0,\dots,n+1,$$

is satisfied, where $\epsilon=1$ or $-1$. This theorem was proved by P.L. Chebyshev in 1854 (cf. [1]) in a more general form, namely for the best uniform approximation of functions by rational functions with fixed degrees of the numerator and denominator. Chebyshev's theorem remains valid if instead of algebraic polynomials one considers polynomials

$$P_n(x)=\sum_{k=0}^nc_k\phi_k(x),$$

where $\{\phi_k(x)\}_{k=0}^n$ is a Chebyshev system. The criterion formulated in Chebyshev's theorem leads to methods for the approximate construction of polynomials of best uniform (Chebyshev) approximation. In a somewhat different formulation Chebyshev's theorem can be extended to functions of a complex variable (cf. [2]) and to abstract functions (cf. [3]).

References

[1] P.L. Chebyshev, "Questions on smallest quantities connected with the approximate representation of functions (1859)" , Collected works , 2 , Moscow-Leningrad (1947) pp. 151–238 (In Russian)
[2] A.N. Kolmogorov, "A remark on the polynomials of Chebyshev deviating the least from a given function" Uspekhi Mat. Nauk , 3 : 1 (1948) pp. 216–221 (In Russian)
[3] S.I. Zukhovitskii, S.B. Stechkin, "On the approximation of abstract functions with values in a Banach space" Dokl. Akad. Nauk SSSR , 106 : 5 (1956) pp. 773–776 (In Russian)
[4] V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)


Comments

References

[a1] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2
How to Cite This Entry:
Chebyshev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_theorem&oldid=15464
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article