The integrability of generalized Garrett-Stanojević sums

Abstract

In this paper we have defined the generalized Garrett-Stanojević cosine sums \[ h n ( x ) = ∑ p = 0 n S p r − 1 Δ r a p {h_n}\left ( x \right ) = \sum \limits _{p = 0}^n {S_p^{r - 1}{\Delta ^r}{a_p}} \] and have proved that under suitable conditions h n → h {h_n} \to h in the L 1 {L^1} -norm, where h ( x ) = a 0 / 2 + ∑ n = 1 ∞ a n cos ⁡ n x h\left ( x \right ) = {a_0}/2 + \sum \nolimits _{n = 1}^\infty {{a_n}\cos nx} . If r = 1 r = 1 , then h n ( x ) {h_n}\left ( x \right ) reduces to the modified cosine sums introduced by Rees and Stanojević.