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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On $ L\sp{1}$ convergence of certain cosine sums


Authors: John W. Garrett and Časlav V. Stanojević
Journal: Proc. Amer. Math. Soc. 54 (1976), 101-105
DOI: https://doi.org/10.1090/S0002-9939-1976-0394002-8
MathSciNet review: 0394002
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Abstract: Rees and Stanojević introduced a new class of modified cosine sums $ \{ {g_n}(x) = \tfrac{1} {2}\sum\nolimits_{k = 0}^n {\Delta a(k) + \sum\nolimits_{k = 1}^n {\sum\nolimits_{j = k}^n {\Delta a(j)\cos kx\} } } } $ and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series $ f$ with coefficients of bounded variation, a Rees-Stanojević cosine sum $ {g_n}$ can be associated such that $ {g_n}$ converges to $ f$ pointwise, and a necessary and sufficient condition for $ {L^1}$ convergence of $ {g_n}$ to $ f$ is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0394002-8
Keywords: $ {L^1}$ convergence of cosine sums
Article copyright: © Copyright 1976 American Mathematical Society