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Difference between revisions of "De la Vallée-Poussin criterion"

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''for pointwise convergence of a Fourier series''
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''for the convergence of Fourier series''
  
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If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302301.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302302.png" /> which is integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302303.png" /> is such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302304.png" /> defined by
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{{MSC|42A20}}
  
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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/d030230/d0302305.png" /></td> </tr></table>
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{{TEX|done}}
  
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302306.png" />, is of bounded variation on some segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302307.png" />, then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302308.png" /> converges at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030230/d0302309.png" /> to the number
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A criterion first proved by De la Vallée-Poussin for the convergence of Fourier series in {{Cite|De}}.  
  
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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/d030230/d03023010.png" /></td> </tr></table>
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'''Theorem'''
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Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function
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\[
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F (t) := \frac{1}{t} \int_0^t \left(f(x+u)+f(x-u) - 2 f(x)\right)\, du \qquad \mbox{for } t>0
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\]
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and $F(0)=0$. If $F$ has bounded variation on some interval $[0, \delta]$ with $\delta>0$, then the Fourier series of $f$ converges to $f(x)$ at $x$.
  
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The de la Vallée-Poussin criterion is stronger than the [[Dini criterion|Dini criterion]], the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]], and the [[Jordan criterion|Jordan criterion]]. It was demonstrated by Ch.J. de la Vallée-Poussin [[#References|[1]]].
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The de la Vallée-Poussin criterion is stronger than the [[Dini criterion|Dini criterion]], the [[Dirichlet theorem|Dirichlet criterion]], and the [[Jordan criterion|Jordan criterion]], it is weaker than the [[Lebesgue criterion]] and not comparable to the [[Young criterion]]. Cp. with Section 3 of chapter III in Volume 1 of {{Cite|Ba}}.  
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Un nouveau cas de convergence des séries de Fourier"  ''Rend. Circ. Mat. Palermo'' , '''31'''  (1911)  pp. 296–299</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon (1964)  (Translated from Russian)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ba}}|| N.K. Bary,  "A treatise on trigonometric series" , Pergamon, 1964.
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|-
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|valign="top"|{{Ref|De}}||  Ch.J. de la Vallée-Poussin,  "Un nouveau cas de convergence des séries de Fourier"  ''Rend. Circ. Mat. Palermo'' , '''31'''  (1911)  pp. 296–299.
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|-
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|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
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|-
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|valign="top"|{{Ref|Zy}}|| A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ.  Press (1988)  {{MR|0933759}}  {{ZBL|0628.42001}}
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|-
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|}

Latest revision as of 20:45, 16 October 2012

for the convergence of Fourier series

2020 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion first proved by De la Vallée-Poussin for the convergence of Fourier series in [De].

Theorem Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function \[ F (t) := \frac{1}{t} \int_0^t \left(f(x+u)+f(x-u) - 2 f(x)\right)\, du \qquad \mbox{for } t>0 \] and $F(0)=0$. If $F$ has bounded variation on some interval $[0, \delta]$ with $\delta>0$, then the Fourier series of $f$ converges to $f(x)$ at $x$.


The de la Vallée-Poussin criterion is stronger than the Dini criterion, the Dirichlet criterion, and the Jordan criterion, it is weaker than the Lebesgue criterion and not comparable to the Young criterion. Cp. with Section 3 of chapter III in Volume 1 of [Ba].

References

[Ba] N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[De] Ch.J. de la Vallée-Poussin, "Un nouveau cas de convergence des séries de Fourier" Rend. Circ. Mat. Palermo , 31 (1911) pp. 296–299.
[Ed] R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Zy] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
How to Cite This Entry:
De la Vallée-Poussin criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_criterion&oldid=22323
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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