View source for De la Vallée-Poussin summation method

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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 [[#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

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

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|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Gronwall,   "Ueber einige Summationsmethoden und ihre Anwendung auf die Fouriersche Reihe"  ''J. Reine Angew. Math.'' , '''147'''  (1917)  pp. 16–35</TD></TR></table>