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A polynomial deviating least from a given function. More precisely, let a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116201.png" /> be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116202.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116203.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116204.png" /> be the set of algebraic polynomials of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116205.png" />. The quantity
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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/a/a011/a011620/a0116206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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is called the [[Best approximation|best approximation]], while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116207.png" />. Polynomials which deviate least from a given continuous function in the uniform metric (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116208.png" />) were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 [[#References|[1]]]. The existence of algebraic polynomials of best approximation was established by E. Borel [[#References|[2]]]. Chebyshev proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a0116209.png" /> is an algebraic polynomial of best approximation in the uniform metric if and only if [[Chebyshev alternation|Chebyshev alternation]] occurs in the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162010.png" />; in this case such a polynomial is unique. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162011.png" />, the algebraic polynomial of best approximation is unique due to the strict convexity of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162013.png" />, it is not unique, but it has been shown by D. Jackson [[#References|[3]]] to be unique for continuous functions. The rate of convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162014.png" /> to zero is given by Jackson's theorems (cf. [[Jackson theorem|Jackson theorem]]).
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A polynomial deviating least from a given function. More precisely, let a measurable function  $  f(x) $
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be in L _ {p} [a, b] $(
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$  p \geq  1 $)  
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and let  $  H _ {n} $
 +
be the set of algebraic polynomials of degree not exceeding  $  n $.  
 +
The quantity
  
In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162015.png" />. If the number of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162016.png" />, an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.
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$$ \tag{* }
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E _ {n} (f) _ {p}  =  \inf _ {P _ {n} (x) \in
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H _ {n} }  \| f (x) - P _ {n} (x) \| _ {L _ {p}  [ a , b ] }
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$$
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 +
is called the [[Best approximation|best approximation]], while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in  $  L _ {p} [a, b] $.
 +
Polynomials which deviate least from a given continuous function in the uniform metric ( $  p = \infty $)
 +
were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 [[#References|[1]]]. The existence of algebraic polynomials of best approximation was established by E. Borel [[#References|[2]]]. Chebyshev proved that  $  P _ {n}  ^ {0} (x) $
 +
is an algebraic polynomial of best approximation in the uniform metric if and only if [[Chebyshev alternation|Chebyshev alternation]] occurs in the difference  $  f(x) - P _ {n}  ^ {0} (x) $;
 +
in this case such a polynomial is unique. If  $  p > 1 $,
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the algebraic polynomial of best approximation is unique due to the strict convexity of the space  $  L _ {p} $.
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If  $  p = 1 $,
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it is not unique, but it has been shown by D. Jackson [[#References|[3]]] to be unique for continuous functions. The rate of convergence of  $  E _ {n} {(f) } _ {p} $
 +
to zero is given by Jackson's theorems (cf. [[Jackson theorem|Jackson theorem]]).
 +
 
 +
In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say $  m $.  
 +
If the number of variables $  m \geq  2 $,  
 +
an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  "Questions on smallest quantities connected with the approximate representation of functions (1859)" , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 478; 152–236  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Borel,  "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Jackson,  "A general class of problems in approximation"  ''Amer. J. Math.'' , '''46'''  (1924)  pp. 215–234</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.L. Garkavi,  "The theory of approximation in normed linear spaces"  ''Itogi Nauk. Mat. Anal. 1967''  (1969)  pp. 75–132  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  "Questions on smallest quantities connected with the approximate representation of functions (1859)" , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 478; 152–236  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Borel,  "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Jackson,  "A general class of problems in approximation"  ''Amer. J. Math.'' , '''46'''  (1924)  pp. 215–234</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.L. Garkavi,  "The theory of approximation in normed linear spaces"  ''Itogi Nauk. Mat. Anal. 1967''  (1969)  pp. 75–132  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Instead of the long phrase  "algebraic polynomial of best approximation"  one also uses the shorter phrase  "best algebraic approximationbest algebraic approximation" , which is not to be confused with the phrase  "best approximation"  for the least error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011620/a01162017.png" />.
+
Instead of the long phrase  "algebraic polynomial of best approximation"  one also uses the shorter phrase  "best algebraic approximationbest algebraic approximation" , which is not to be confused with the phrase  "best approximation"  for the least error $  E _ {n} (f) _ {p} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Meinardus,  "Approximation von Funktionen und ihre numerische Behandlung" , Springer  (1964)  pp. Chapt. 1, §5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Meinardus,  "Approximation von Funktionen und ihre numerische Behandlung" , Springer  (1964)  pp. Chapt. 1, §5</TD></TR></table>

Latest revision as of 16:10, 1 April 2020


A polynomial deviating least from a given function. More precisely, let a measurable function $ f(x) $ be in $ L _ {p} [a, b] $( $ p \geq 1 $) and let $ H _ {n} $ be the set of algebraic polynomials of degree not exceeding $ n $. The quantity

$$ \tag{* } E _ {n} (f) _ {p} = \inf _ {P _ {n} (x) \in H _ {n} } \| f (x) - P _ {n} (x) \| _ {L _ {p} [ a , b ] } $$

is called the best approximation, while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in $ L _ {p} [a, b] $. Polynomials which deviate least from a given continuous function in the uniform metric ( $ p = \infty $) were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 [1]. The existence of algebraic polynomials of best approximation was established by E. Borel [2]. Chebyshev proved that $ P _ {n} ^ {0} (x) $ is an algebraic polynomial of best approximation in the uniform metric if and only if Chebyshev alternation occurs in the difference $ f(x) - P _ {n} ^ {0} (x) $; in this case such a polynomial is unique. If $ p > 1 $, the algebraic polynomial of best approximation is unique due to the strict convexity of the space $ L _ {p} $. If $ p = 1 $, it is not unique, but it has been shown by D. Jackson [3] to be unique for continuous functions. The rate of convergence of $ E _ {n} {(f) } _ {p} $ to zero is given by Jackson's theorems (cf. Jackson theorem).

In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say $ m $. If the number of variables $ m \geq 2 $, an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.

References

[1] P.L. Chebyshev, "Questions on smallest quantities connected with the approximate representation of functions (1859)" , Collected works , 2 , Moscow-Leningrad (1947) pp. 478; 152–236 (In Russian)
[2] E. Borel, "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars (1905)
[3] D. Jackson, "A general class of problems in approximation" Amer. J. Math. , 46 (1924) pp. 215–234
[4] A.L. Garkavi, "The theory of approximation in normed linear spaces" Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 (In Russian)

Comments

Instead of the long phrase "algebraic polynomial of best approximation" one also uses the shorter phrase "best algebraic approximationbest algebraic approximation" , which is not to be confused with the phrase "best approximation" for the least error $ E _ {n} (f) _ {p} $.

References

[a1] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[a2] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
[a3] G.W. Meinardus, "Approximation von Funktionen und ihre numerische Behandlung" , Springer (1964) pp. Chapt. 1, §5
How to Cite This Entry:
Algebraic polynomial of best approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_polynomial_of_best_approximation&oldid=45064
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article