Young criterion

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One of the sufficiency criteria for the convergence of a Fourier series at a point. Let a function have period , be Lebesgue integrable over the interval and, on putting , let it satisfy at the point the conditions: 1) as ; 2) the function is of finite variation on the interval , , where is some fixed number; and 3) as . Then the Fourier series of at converges to (cf. [2]). Young's criterion is stronger than the Jordan criterion. It was established by W.H. Young [1].


[1] W.H. Young, "On the convergence of the derived series of Fourier series" Proc. London Math. Soc. , 17 (1916) pp. 195–236
[2] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)



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