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A function
 
A 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/l/l057/l057840/l0578401.png" /></td> </tr></table>
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$$
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L _ {n}  ^  \Phi  ( t)  = \int\limits _ { a } ^ { b }
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\left | \sum _ { k=1 } ^ { n }  \phi _ {k} ( x) \phi _ {k} ( t) \right |  d x ,\  t \in [ a , b ] ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578402.png" /> is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578404.png" />. Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has
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where $  \Phi = \{ \phi _ {k} \} _ {k=1}  ^  \infty  $
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is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval $  [ a , b ] $,
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$  n = 1 , 2 , . . . $.  
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Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has
  
<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/l/l057/l057840/l0578405.png" /></td> </tr></table>
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$$
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L _ {n}  ^  \Phi  ( t)  = \
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\sup _ {f : \| f \| _ {C [ a , b ] }
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\leq  1 }  | S _ {n} ( f  ) | ,\ \
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t \in [ a , b ] ,
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$$
  
 
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/l/l057/l057840/l0578406.png" /></td> </tr></table>
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$$
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S _ {n} ( f  ) ( t)  = \sum _ { k=1 } ^ { n }
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c _ {k} ( f  ) \phi _ {k} ( t)
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$$
  
is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578407.png" />-th partial sum of the [[Fourier series|Fourier series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578408.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l0578409.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057840/l05784010.png" /> is the [[Trigonometric system|trigonometric system]], the Lebesgue functions are constant and reduce to the [[Lebesgue constants|Lebesgue constants]]. They were introduced by H. Lebesgue.
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is the $  n $-th partial sum of the [[Fourier series|Fourier series]] of $  f $
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with respect to $  \Phi $.  
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In the case when $  \Phi $
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is the [[Trigonometric system|trigonometric system]], the Lebesgue functions are constant and reduce to the [[Lebesgue constants|Lebesgue constants]]. They were introduced by H. Lebesgue.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>

Latest revision as of 19:29, 28 February 2021


A function

$$ L _ {n} ^ \Phi ( t) = \int\limits _ { a } ^ { b } \left | \sum _ { k=1 } ^ { n } \phi _ {k} ( x) \phi _ {k} ( t) \right | d x ,\ t \in [ a , b ] , $$

where $ \Phi = \{ \phi _ {k} \} _ {k=1} ^ \infty $ is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval $ [ a , b ] $, $ n = 1 , 2 , . . . $. Lebesgue functions are defined similarly in the case when an orthonormal system is specified on an arbitrary measure space. One has

$$ L _ {n} ^ \Phi ( t) = \ \sup _ {f : \| f \| _ {C [ a , b ] } \leq 1 } | S _ {n} ( f ) | ,\ \ t \in [ a , b ] , $$

where

$$ S _ {n} ( f ) ( t) = \sum _ { k=1 } ^ { n } c _ {k} ( f ) \phi _ {k} ( t) $$

is the $ n $-th partial sum of the Fourier series of $ f $ with respect to $ \Phi $. In the case when $ \Phi $ is the trigonometric system, the Lebesgue functions are constant and reduce to the Lebesgue constants. They were introduced by H. Lebesgue.

References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
How to Cite This Entry:
Lebesgue function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_function&oldid=13866
This article was adapted from an original article by B.S. Kashin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article