# Jordan criterion

From Encyclopedia of Mathematics

*for the convergence of Fourier series*

If a -periodic function has bounded variation on the interval , then its Fourier series converges at every point to ; if, in addition, is continuous on , then its Fourier series converges to it uniformly on every interval strictly inside . The criterion was established by C. Jordan [1]; it generalizes the Dirichlet theorem on the convergence of Fourier series of piecewise-monotone functions.

#### References

[1] | C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230 |

[2] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |

#### Comments

The Jordan criterion is also called the Dirichlet–Jordan test, cf. [a1].

#### References

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

**How to Cite This Entry:**

Jordan criterion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Jordan_criterion&oldid=17285

This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article