Lebesgue criterion

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A criterion for pointwise convergence of Fourier series. If a -periodic function , integrable on the interval , satisfies the condition


at a point for some , where

then the Fourier series of at converges to the number . The criterion was proved by H. Lebesgue [1]. Condition (*) is equivalent to the aggregate of the two conditions

The Lebesgue criterion is more powerful then the Dirichlet criterion (convergence of series); the Jordan criterion; the Dini criterion; the de la Vallée-Poussin criterion; and the Young criterion.


[1] H. Lebesgue, "Récherches sur le convergence des séries de Fourier" Math. Ann. , 61 (1905) pp. 251–280
[2] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)



[a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
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Lebesgue criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article