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A criterion for pointwise convergence of [[Fourier series|Fourier series]]. If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578101.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578102.png" />, integrable on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578103.png" />, satisfies the condition
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''for the convergence of Fourier 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/l/l057/l057810/l0578104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{MSC|42A20}}
  
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at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578105.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578106.png" />, where
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{{TEX|done}}
  
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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/l/l057/l057810/l0578107.png" /></td> </tr></table>
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A criterion first proved by Lebesgue for the convergence of Fourier series in {{Cite|Le}}.  
  
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then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578108.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l0578109.png" /> converges to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l05781010.png" />. The criterion was proved by H. Lebesgue [[#References|[1]]]. Condition (*) is equivalent to the aggregate of the two conditions
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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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<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/l/l057/l057810/l05781011.png" /></td> </tr></table>
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\[
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\varphi (u):= f(x+u)+f(x-u) - 2 f(x)
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l05781012.png" /></td> </tr></table>
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\]
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If there is $\delta>0$ such that
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The Lebesgue criterion is more powerful then the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]]; the [[Jordan criterion|Jordan criterion]]; the [[Dini criterion|Dini criterion]]; the [[De la Vallée-Poussin criterion|de la Vallée-Poussin criterion]]; and the [[Young criterion|Young criterion]].
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\[
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\lim_{h\downarrow 0} \int_h^\delta \left|\frac{\varphi (u+h)}{u+h} - \frac{\varphi (u)}{u}\right|\, du \;=\; 0
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====References====
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\]
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Récherches sur le convergence des séries de Fourier"  ''Math. Ann.'' , '''61'''  (1905) pp. 251–280</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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then the Fourier series of $f$ converges to $f(x)$ at $x$.
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====Comments====
 
  
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Cp. with Section 6 of Chapter III in volume 1 of {{Cite|Ba}} and Section 11 of Chapter II in volume 1 of {{Cite|Zy}}. The Lebesgue criterion is stronger then the [[Dirichlet theorem|Dirichlet criterion]], the [[Jordan criterion|Jordan criterion]], the [[Dini criterion|Dini criterion]], the [[De la Vallée-Poussin criterion|de la Vallée-Poussin criterion]], and the [[Young criterion|Young criterion]]. Cp. with Section 7 of Chapter III in volume 1 of {{Cite|Ba}}.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</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|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
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|-
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|valign="top"|{{Ref|Le}}|| H. Lebesgue,  "Récherches sur le convergence des séries de Fourier"  ''Math. Ann.'' , '''61''' (1905)  pp. 251–280.
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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 20:37, 16 October 2012

for the convergence of Fourier series

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

A criterion first proved by Lebesgue for the convergence of Fourier series in [Le].

Theorem Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function \[ \varphi (u):= f(x+u)+f(x-u) - 2 f(x) \] If there is $\delta>0$ such that \[ \lim_{h\downarrow 0} \int_h^\delta \left|\frac{\varphi (u+h)}{u+h} - \frac{\varphi (u)}{u}\right|\, du \;=\; 0 \] then the Fourier series of $f$ converges to $f(x)$ at $x$.

Cp. with Section 6 of Chapter III in volume 1 of [Ba] and Section 11 of Chapter II in volume 1 of [Zy]. The Lebesgue criterion is stronger then the Dirichlet criterion, the Jordan criterion, the Dini criterion, the de la Vallée-Poussin criterion, and the Young criterion. Cp. with Section 7 of Chapter III in volume 1 of [Ba].

References

[Ba] N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[Ed] R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Le] H. Lebesgue, "Récherches sur le convergence des séries de Fourier" Math. Ann. , 61 (1905) pp. 251–280.
[Zy] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
How to Cite This Entry:
Lebesgue criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_criterion&oldid=28449
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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