On the Uniform Convergence of Sine Series with Square Root

Abstract EN

Chaundy and Jolliffe proved that if {ck}k=1∞ is a nonincreasing real sequence with limk→∞ck=0, then the series ∑k=1∞cksinkx converges uniformly if and only if kck→0.

The purpose of this paper is to show that kck→0 is a necessary and sufficient condition for the uniform convergence of series ∑k=1∞cksinkθ in θ∈[0,π].

However for ∑k=1∞cksink2θ it is not true in θ∈[0,π].