On the Uniform Convergence of Sine Series with Square Root

Abstract

Chaundy and Jolliffe proved that if {ck}k=1∞ is a nonincreasing real sequence with limk→∞ck=0, then the series ∑k=1∞‍cksin⁡kx 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∞‍cksin⁡kθ in θ∈[0,π]. However for ∑k=1∞‍cksin⁡k2θ it is not true in θ∈[0,π].

Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

“…The cases α = 1/2 and α = 2 for the series (1.1) were considered in [8], where it was shown that the condition c k k → 0 is necessary and sufficient for the series (1.1) to converge uniformly on the interval [0, π] for α = 1/2, and for α = 2 a necessary and sufficient condition is ∞ k=1 c k < ∞. We obtain the following.…”

A Note on Derivative of Sine Series with Square Root

“…Let f ðxÞ = ∑ n=4 ðsin ðx ffiffiffi n p Þ/n ln nÞ. It converges uniformly on ½0, π[5]. We show that f is not differentiable at x = 0.…”

“…In the recent paper [5], it was proved that ∑a n sin ð ffiffiffi n p xÞ converges uniformly on ½0, π if and only if na n ⟶ 0.…”