Application of approximation theory for sine series with coefficient from class supremum bounded variation sequences of first type

The coefficients of sine series which is monotone sequence (MS) have been generalized to General Monotone Sequences (GMS), Group Bounded Variation Sequences (GBVS), Mean Value Bounded Variation Sequences (MVBVS) and Supremum Bounded Variation Sequences of the first type (SBVS) and has been proven that MS ⊊ GMS ⊊ GBVS ⊊ MV BVS. Further it has also been shown that MBVS ⊊ S BVS. Furthermore, in this paper it will be discuss the application of the approximation theory of the sine sequence in the class of supremum Bounded Variation Sequences of the first type which is the generalization of the monotone sequences class.