# Lebesgue criterion

From Encyclopedia of Mathematics

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.

#### References

[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) |

#### Comments

#### References

[a1] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |

**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