Ultimate Generalization to Monotonicity for Uniform Convergence of Trigonometric Series

2006/11/27 by Song-Ping Zhou, Songping Zhou, Ping Zhou +5
Mathematics · #42A20 #42A32 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Analysis #math.CA #msc:42A20 #msc:42A32

paper · pdf · doi:10.48550/arxiv.math/0611805

21 pages

arxiv created 2006/11/27 · openalex publication_date 2006/11/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_nsin 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.

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