Jordan criterion

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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.


[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)


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


[a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Jordan 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