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.

We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from one sided to two sided, while some important classical results keep true.

Abstract. We first give a necessary and sufficient condition forunder the condition that {λ n } (where λ n is a n or b n respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λ n and the sum function φ(x) under the condition that {λ n } ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of φ(x) in L p norm.