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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
| + | ''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>
| + | {{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
| + | {{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>
| + | 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
| + | '''Theorem''' |
− | | + | Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function |
− | <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>
| + | \[ |
− | | + | \varphi (u):= f(x+u)+f(x-u) - 2 f(x) |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057810/l05781012.png" /></td> </tr></table>
| + | \] |
− | | + | If there is $\delta>0$ such that |
− | 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]].
| + | \[ |
− | | + | \lim_{h\downarrow 0} \int_h^\delta \left|\frac{\varphi (u+h)}{u+h} - \frac{\varphi (u)}{u}\right|\, du \;=\; 0 |
− | ====References====
| + | \] |
− | <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>
| + | then the Fourier series of $f$ converges to $f(x)$ at $x$. |
− | | |
− | | |
− | | |
− | ====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==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988)</TD></TR></table>
| + | {| |
| + | |- |
| + | |valign="top"|{{Ref|Ba}}|| N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964. |
| + | |- |
| + | |valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967. |
| + | |- |
| + | |valign="top"|{{Ref|Le}}|| H. Lebesgue, "Récherches sur le convergence des séries de Fourier" ''Math. Ann.'' , '''61''' (1905) pp. 251–280. |
| + | |- |
| + | |valign="top"|{{Ref|Zy}}|| A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) {{MR|0933759}} {{ZBL|0628.42001}} |
| + | |- |
| + | |} |
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
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How to Cite This Entry:
Lebesgue criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_criterion&oldid=17261
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article