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Difference between revisions of "De la Vallée-Poussin summation method"

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A method for summing series of numbers. It is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030280/d0302801.png" />. A series
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<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/d/d030/d030280/d0302802.png" /></td> </tr></table>
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has a de la Vallée-Poussin sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030280/d0302803.png" /> if the relation
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A method for summing series of numbers. It is denoted by the symbol  $  ( VP) $.  
 +
A series
  
<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/d/d030/d030280/d0302804.png" /></td> </tr></table>
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$$
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\sum _ {k = 0 } ^  \infty  a _ {k}  $$
  
<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/d/d030/d030280/d0302805.png" /></td> </tr></table>
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has a de la Vallée-Poussin sum  $  S $
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if the relation
  
is valid. The method was proposed by Ch.J. de la Vallée-Poussin [[#References|[1]]]. For the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030280/d0302806.png" /> the de la Vallée-Poussin averages (see also [[De la Vallée-Poussin singular integral|de la Vallée-Poussin singular integral]]) are of the form
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$$
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\lim\limits _ {n \rightarrow \infty }
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\left [ a _ {0} +
  
<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/d/d030/d030280/d0302807.png" /></td> </tr></table>
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\frac{n}{n + 1 }
 +
a _ {1} +
 +
 
 +
\frac{n ( n - 1) }{( n + 1) ( n + 2) }
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 +
a _ {2} + \right . \cdots
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$$
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$$
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\cdots \left .
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+ {n! \over ( n + 1) \cdots 2n } a _ {n} \right ]  = S
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$$
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is valid. The method was proposed by Ch.J. de la Vallée-Poussin [[#References|[1]]]. For the Fourier series of a function  $  f \in L [ 0, 2 \pi ] $
 +
the de la Vallée-Poussin averages (see also [[De la Vallée-Poussin singular integral|de la Vallée-Poussin singular integral]]) are of the form
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$$
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V _ {n} ( f, x)  = \
 +
\int\limits _ {- \pi } ^  \pi 
 +
f ( x + t) \tau _ {n} ( t)  dt,
 +
$$
  
 
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/d/d030/d030280/d0302808.png" /></td> </tr></table>
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$$
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\tau _ {n} ( t)  = \
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 +
\frac{1}{2 \pi }
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 +
\frac{2  ^ {2n} ( n!)  ^ {2} }{( 2n)! }
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\
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\cos  ^ {2n}  {
 +
\frac{t}{2}
 +
}
 +
$$
  
 
is the so-called de la Vallée-Poussin kernel. The de la Vallée-Poussin summation method is a regular summation method (cf. [[Regular summation methods|Regular summation methods]]). The method is stronger than all [[Cesàro summation methods|Cesàro summation methods]] (cf. [[Inclusion of summation methods|Inclusion of summation methods]]). In view of its weak approximative properties, the de la Vallée-Poussin summation method is practically never used in the theory of approximation of functions.
 
is the so-called de la Vallée-Poussin kernel. The de la Vallée-Poussin summation method is a regular summation method (cf. [[Regular summation methods|Regular summation methods]]). The method is stronger than all [[Cesàro summation methods|Cesàro summation methods]] (cf. [[Inclusion of summation methods|Inclusion of summation methods]]). In view of its weak approximative properties, the de la Vallée-Poussin summation method is practically never used in the theory of approximation of functions.

Latest revision as of 07:46, 13 May 2022


A method for summing series of numbers. It is denoted by the symbol $ ( VP) $. A series

$$ \sum _ {k = 0 } ^ \infty a _ {k} $$

has a de la Vallée-Poussin sum $ S $ if the relation

$$ \lim\limits _ {n \rightarrow \infty } \left [ a _ {0} + \frac{n}{n + 1 } a _ {1} + \frac{n ( n - 1) }{( n + 1) ( n + 2) } a _ {2} + \right . \cdots $$

$$ \cdots \left . + {n! \over ( n + 1) \cdots 2n } a _ {n} \right ] = S $$

is valid. The method was proposed by Ch.J. de la Vallée-Poussin [1]. For the Fourier series of a function $ f \in L [ 0, 2 \pi ] $ the de la Vallée-Poussin averages (see also de la Vallée-Poussin singular integral) are of the form

$$ V _ {n} ( f, x) = \ \int\limits _ {- \pi } ^ \pi f ( x + t) \tau _ {n} ( t) dt, $$

where

$$ \tau _ {n} ( t) = \ \frac{1}{2 \pi } \frac{2 ^ {2n} ( n!) ^ {2} }{( 2n)! } \ \cos ^ {2n} { \frac{t}{2} } $$

is the so-called de la Vallée-Poussin kernel. The de la Vallée-Poussin summation method is a regular summation method (cf. Regular summation methods). The method is stronger than all Cesàro summation methods (cf. Inclusion of summation methods). In view of its weak approximative properties, the de la Vallée-Poussin summation method is practically never used in the theory of approximation of functions.

References

[1] Ch.J. de la Vallée-Poussin, "Sur l'approximation des fonctions d'une variable reélle et de leurs dérivées par des polynômes et des suites limitées de Fourier" Bull. Acad. Belg. , 3 (1908) pp. 193–254
[2] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[3] T. Gronwall, "Ueber einige Summationsmethoden und ihre Anwendung auf die Fouriersche Reihe" J. Reine Angew. Math. , 147 (1917) pp. 16–35
How to Cite This Entry:
De la Vallée-Poussin summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_summation_method&oldid=22333
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article