Ultimate generalization to monotonicity for uniform convergence of trigonometric series

Abstract

Chaundy and Jolliffe [4] 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 ∞ 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 and Jolliffe theorem in the complex space.2000 Mathematics Subject Classification. 42A20 42A32.

On the uniform convergence of double sine series

“…We note that in [11] Theorem B was proved in the following more general form: If {a k } ⊂ R + belongs to the class MVBVS, then condition (1.2) is necessary and sufficient for the uniform convergence of series (1.1), and also for the continuity of its sum function. The addendum on the continuity of the sum function is a variant of a theorem of Paley [8] with almost the same proof.…”

“…A sequence {a k } ⊂ R + is said to belong to the class MVBVS (= mean value bounded variation sequences) if there exist constants C and λ ≥ 2, both depending only on the sequence {a k }, such that The following theorem was proved in [11,Theorem 5].…”

“…An analysis of the proofs in [9] and [11] reveals that the sufficiency part in Theorem B as well as Theorem C remain valid if we consider sequences {c k : k = 1, 2, . .…”

“…In Section 1, we have noted that the first statement in Theorem C has not been explicity proved in [11]. Instead, it is only indicated there that the proof is similar to that of [11,Proposition 3].…”

“…Instead, it is only indicated there that the proof is similar to that of [11,Proposition 3]. For the reader's convenience, below we present a proof of the more general Theorem C .…”