Jordan criterion

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)