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

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m (moved De la Vallee-Poussin criterion to De la Vallée-Poussin criterion over redirect: accented title)
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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 Jordan 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]]. 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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|}

Revision as of 20:00, 16 October 2012

for the convergence of Fourier series

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

A criterion first proved by Jordan 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. 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=23242
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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