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On the uniform convergence of sine series with quasimonotone coefficients. (English) Zbl 0756.42006
Summary: In this note, the well theorem of T. W. Chaundy and A. E. Jolliffe [Proc. Lond. Math. Soc., II. Ser. 15, 214-216 (1916)] giving necessary and sufficient conditions for the uniform convergence of a sine series with decreasing coefficients is extended to the case where the coefficients are quasi-monotone. A related theorem of Hardy is similarly extended. These results are, in a certain sense, best possible.

##### MSC:
 42A20 Convergence and absolute convergence of Fourier and trigonometric series
Full Text:
##### References:
 [1] Chaundy, T.W; Jolliffe, A.E, The uniform convergence of a certain class of trigonometrical series, (), 214-216 · JFM 46.0455.03 [2] Clunie, J, An extension of quasimonotone series, Math. student, 20, 107-112, (1952) [3] Hardy, G.H, Some theorems concerning trigonometrical series of a special type, (), 441-448 · Zbl 0002.25403 [4] Hardy, G.H; Rogosinski, W.W, Notes on Fourier series (I): on sine series with positive coefficients, J. London math. soc., 18, 50-57, (1943) · Zbl 0063.01927 [5] Jolliffe, A.E, On certain trigonometrical series which have a necessary and sufficient condition for uniform convergence, (), 191-195 · JFM 47.0914.03
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