On the Uniform Convergence of Sine Series with Square Root

. Chaundy and Jolliffe proved that if {𝑐 𝑘 } ∞𝑘=1 is a nonincreasing real sequence with lim 𝑘󳨀→∞ 𝑐 𝑘 = 0 , then the series ∑ ∞𝑘=1 c 𝑘 sin 𝑘𝑥 converges uniformly if and only if 𝑘𝑐 𝑘 󳨀→ 0 . The purpose of this paper is to show that 𝑘𝑐 𝑘 󳨀→ 0 is a necessary and sufficient condition for the uniform convergence of series ∑ ∞𝑘=1 c 𝑘 sin √𝑘𝜃 in 𝜃 ∈ [0,𝜋] . However for ∑ ∞𝑘=1 c 𝑘 sin 𝑘 2 𝜃 it is not true in 𝜃 ∈ [0,𝜋] .

Theorem 4 was generalized by Kórus [6].He has defined new classes of double sequences ( 1 ) to obtain those generalizations.
Duzinkiewicz and Szal [7] introduce a new class of double sequence called (, , , ), which is a generalization of the class considered by Kórus, and they obtain sufficient and necessary conditions for uniform convergence of double sine series.
A series was motivation for the generalization of the Theorem 1.Such series were studied by Paley and Wiener who called them nonharmonic Fourier series.They proved the following [8].We will consider a special case of the series (1) for   = √  and   =  2 ,  ≥ 1, which does not meet the assumptions of the above theorem.
This follows from (6).Note that the following condition is fulfilled: In view of (8) the following inequality is satisfied for Thus the sequence   is increasing with respect to  and lim This follows from (9).Finally for  ≤  and  ≥ √, This follows from ( 7), (10).
To prove the case  ≤ √ we first observe the following.