Ultimate generalization to monotonicity for uniform convergence of trigonometric series

By Songping Zhou, P. Zhou, Dansheng Yu. 2006

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Chaundy and Jolliffe proved that if {an} is a non-increasing (monotonic) real sequence with limn→∞an = 0, then a necessary and sufficient condition for the uniform convergence of the series Σn=1∞an sin nx is limn→∞nan = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {an} 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c5ee6ee938280daeaee68bf2c50167b60560a61f