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Difference between revisions of "Lebesgue function"

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$$  
 
$$  
 
L _ {n}  ^  \Phi  ( t)  =  \int\limits _ { a } ^ { b }  
 
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 ] ,
+
\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  $
+
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 ] $,  
 
is a given system of functions, orthonormal with respect to the Lebesgue measure on the interval  $  [ a , b ] $,  
 
$  n = 1 , 2 , .  .  . $.  
 
$  n = 1 , 2 , .  .  . $.  
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$$  
 
$$  
S _ {n} ( f  ) ( t)  =  \sum _ { k= } 1 ^ { n }  
+
S _ {n} ( f  ) ( t)  =  \sum _ { k=1 } ^ { n }  
 
c _ {k} ( f  ) \phi _ {k} ( t)
 
c _ {k} ( f  ) \phi _ {k} ( t)
 
$$
 
$$
  
is the  $  n $-
+
is the  $  n $-th partial sum of the [[Fourier series|Fourier series]] of  $  f $
th partial sum of the [[Fourier series|Fourier series]] of  $  f $
 
 
with respect to  $  \Phi $.  
 
with respect to  $  \Phi $.  
 
In the case when  $  \Phi $
 
In the case when  $  \Phi $

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=47601
This article was adapted from an original article by B.S. Kashin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article