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The integrability of generalized Garrett-Stanojević sums

Authors: Niranjan Singh and K. M. Sharma
Journal: Proc. Amer. Math. Soc. 104 (1988), 135-144
MSC: Primary 42A20; Secondary 42A24
MathSciNet review: 929424
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Abstract: In this paper we have defined the generalized Garrett-Stanojević cosine sums

$\displaystyle {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} \to h$ in the $ {L^1}$-norm, where $ h\left( x \right) = {a_0}/2 + \sum\nolimits_{n = 1}^\infty {{a_n}\cos nx} $. If $ r = 1$, then $ {h_n}\left( x \right)$ reduces to the modified cosine sums introduced by Rees and Stanojević.

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Keywords: Cesàro means, $ {L^1}$-convergence, generalized cosine sums
Article copyright: © Copyright 1988 American Mathematical Society

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