View source for De la Vallée-Poussin summation method

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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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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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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where

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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>