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An inequality appearing in a problem of H. Bohr [[#References|[1]]] on the boundedness over the entire real axis of the integral of an almost-periodic function. The ultimate form of this inequality was given by J. Favard [[#References|[2]]]; the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function
 
An inequality appearing in a problem of H. Bohr [[#References|[1]]] on the boundedness over the entire real axis of the integral of an almost-periodic function. The ultimate form of this inequality was given by J. Favard [[#References|[2]]]; the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function
  
<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/b/b016/b016790/b0167901.png" /></td> </tr></table>
+
$$
 +
f(x)  = \
 +
\sum _ { k=n } ^  \infty 
 +
(a _ {k}  \cos  kx + b _ {k}  \sin  kx)
 +
$$
  
with continuous derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016790/b0167902.png" /> for given constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016790/b0167903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016790/b0167904.png" /> which are natural numbers. The accepted form of the Bohr–Favard inequality is
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with continuous derivative $  f  ^ {(r)} (x) $
 +
for given constants $  r $
 +
and $  n $
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which are natural numbers. The accepted form of the Bohr–Favard inequality is
  
<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/b/b016/b016790/b0167905.png" /></td> </tr></table>
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$$
 +
\| f \| _ {C}  \leq  K  \| f  ^ {(r)} \| _ {C} ,
 +
$$
  
<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/b/b016/b016790/b0167906.png" /></td> </tr></table>
+
$$
 +
\| f \| _ {C}  = \max _ {x \in [0, 2 \pi ] }  | f(x) | ,
 +
$$
  
with the best constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016790/b0167907.png" />:
+
with the best constant $  K = K (n, r) $:
  
<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/b/b016/b016790/b0167908.png" /></td> </tr></table>
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$$
 +
= \sup _ {\| f  ^ {(r)} \| _ {C} \leq  1 } \
 +
\| f \| _ {C} .
 +
$$
  
The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016790/b0167909.png" />-th derivative by trigonometric polynomials of an order at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016790/b01679010.png" /> and with the notion of Kolmogorov's width in the class of differentiable functions (cf. [[Width|Width]]).
+
The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its $  r $-
 +
th derivative by trigonometric polynomials of an order at most $  n $
 +
and with the notion of Kolmogorov's width in the class of differentiable functions (cf. [[Width|Width]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,  "Un théorème général sur l'intégration d'un polynôme trigonométrique"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''200'''  (1935)  pp. 1276–1277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Favard,  "Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques"  ''Bull. Sci. Math. (2)'' , '''61'''  (1937)  pp. 209–224; 243–256</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,  "Un théorème général sur l'intégration d'un polynôme trigonométrique"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''200'''  (1935)  pp. 1276–1277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Favard,  "Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques"  ''Bull. Sci. Math. (2)'' , '''61'''  (1937)  pp. 209–224; 243–256</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


An inequality appearing in a problem of H. Bohr [1] on the boundedness over the entire real axis of the integral of an almost-periodic function. The ultimate form of this inequality was given by J. Favard [2]; the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function

$$ f(x) = \ \sum _ { k=n } ^ \infty (a _ {k} \cos kx + b _ {k} \sin kx) $$

with continuous derivative $ f ^ {(r)} (x) $ for given constants $ r $ and $ n $ which are natural numbers. The accepted form of the Bohr–Favard inequality is

$$ \| f \| _ {C} \leq K \| f ^ {(r)} \| _ {C} , $$

$$ \| f \| _ {C} = \max _ {x \in [0, 2 \pi ] } | f(x) | , $$

with the best constant $ K = K (n, r) $:

$$ K = \sup _ {\| f ^ {(r)} \| _ {C} \leq 1 } \ \| f \| _ {C} . $$

The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its $ r $- th derivative by trigonometric polynomials of an order at most $ n $ and with the notion of Kolmogorov's width in the class of differentiable functions (cf. Width).

References

[1] H. Bohr, "Un théorème général sur l'intégration d'un polynôme trigonométrique" C.R. Acad. Sci. Paris Sér. I Math. , 200 (1935) pp. 1276–1277
[2] J. Favard, "Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques" Bull. Sci. Math. (2) , 61 (1937) pp. 209–224; 243–256
[3] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
How to Cite This Entry:
Bohr-Favard inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr-Favard_inequality&oldid=46096
This article was adapted from an original article by L.V. Taikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article