Ultimate generalization to monotonicity for uniform convergence of trigonometric series

Chaundy and Jolliffe proved that if {a _n } is a non-increasing (monotonic) real sequence with lim _n→∞ a _n = 0, then a necessary and sufficient condition for the uniform convergence of the series Σ _n=1 ^∞ a _n sin nx is lim _n→∞ na _n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a _n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.