Difference between revisions of "Lebesgue criterion"
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| − | + | ''for the convergence of Fourier series'' | |
| − | + | {{MSC|42A20}} | |
| − | + | {{TEX|done}} | |
| − | + | A criterion first proved by Lebesgue for the convergence of Fourier series in {{Cite|Le}}. | |
| − | + | '''Theorem''' | |
| − | + | Consider a summable $2\pi$-periodic function $f: \mathbb R \to \mathbb R$, 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 {{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==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 11:59, 14 December 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 $f: \mathbb R \to \mathbb R$, 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=17261
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