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The inequality
 
The inequality
  
<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/f/f038/f038270/f0382701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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\| x \| _ {C [ 0, 2 \pi ] }  \leq  \
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M K _ {r} n  ^ {-} r ,\ \
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r = 1, 2 \dots
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$$
  
 
where
 
where
  
<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/f/f038/f038270/f0382702.png" /></td> </tr></table>
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$$
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K _ {r}  = \
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{
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\frac{4} \pi
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}
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\sum _ {k = 0 } ^  \infty 
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(- 1) ^ {k ( r + 1) }
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( 2k + 1) ^ {- r - 1 } ,
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$$
  
and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038270/f0382703.png" /> is orthogonal to every trigonometric polynomial of order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038270/f0382704.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038270/f0382705.png" /> inequality (*) was proved by H. Bohr (1935), so it is also called the Bohr inequality and the [[Bohr–Favard inequality|Bohr–Favard inequality]]. For an arbitrary positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038270/f0382706.png" /> inequality (*) was proved by J. Favard [[#References|[1]]].
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and the function $  x ( t) \in W  ^ {r} MC $
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is orthogonal to every trigonometric polynomial of order not exceeding $  n - 1 $.  
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For $  r = 1 $
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inequality (*) was proved by H. Bohr (1935), so it is also called the Bohr inequality and the [[Bohr–Favard inequality|Bohr–Favard inequality]]. For an arbitrary positive integer $  r $
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inequality (*) was proved by J. Favard [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Favard,  "Sur l'approximation des fonctions périodiques par des polynomes trigonométriques"  ''C.R. Acad. Sci. Paris'' , '''203'''  (1936)  pp. 1122–1124</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Favard,  "Sur l'approximation des fonctions périodiques par des polynomes trigonométriques"  ''C.R. Acad. Sci. Paris'' , '''203'''  (1936)  pp. 1122–1124</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For a definition of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038270/f0382707.png" /> cf. [[Favard problem|Favard problem]].
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For a definition of the space $  W  ^ {r} MC $
 +
cf. [[Favard problem|Favard problem]].

Revision as of 19:38, 5 June 2020


The inequality

$$ \tag{* } \| x \| _ {C [ 0, 2 \pi ] } \leq \ M K _ {r} n ^ {-} r ,\ \ r = 1, 2 \dots $$

where

$$ K _ {r} = \ { \frac{4} \pi } \sum _ {k = 0 } ^ \infty (- 1) ^ {k ( r + 1) } ( 2k + 1) ^ {- r - 1 } , $$

and the function $ x ( t) \in W ^ {r} MC $ is orthogonal to every trigonometric polynomial of order not exceeding $ n - 1 $. For $ r = 1 $ inequality (*) was proved by H. Bohr (1935), so it is also called the Bohr inequality and the Bohr–Favard inequality. For an arbitrary positive integer $ r $ inequality (*) was proved by J. Favard [1].

References

[1] J. Favard, "Sur l'approximation des fonctions périodiques par des polynomes trigonométriques" C.R. Acad. Sci. Paris , 203 (1936) pp. 1122–1124
[2] V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)

Comments

For a definition of the space $ W ^ {r} MC $ cf. Favard problem.

How to Cite This Entry:
Favard inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_inequality&oldid=46906
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article