Difference between revisions of "Favard inequality"
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The inequality | The inequality | ||
| − | + | $$ \tag{* } | |
| + | \| x \| _ {C [ 0, 2 \pi ] } \leq \ | ||
| + | M K _ {r} n ^ {-} r ,\ \ | ||
| + | r = 1, 2 \dots | ||
| + | $$ | ||
where | where | ||
| − | + | $$ | |
| + | K _ {r} = \ | ||
| + | { | ||
| + | \frac{4} \pi | ||
| + | } | ||
| + | \sum _ {k = 0 } ^ \infty | ||
| + | (- 1) ^ {k ( r + 1) } | ||
| + | ( 2k + 1) ^ {- r - 1 } , | ||
| + | $$ | ||
| − | and the function | + | 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|Bohr–Favard inequality]]. For an arbitrary positive integer $ r $ | ||
| + | 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 | + | 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
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