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A note on the absolute convergence of lacunary Fourier series


Authors: N. V. Patel and V. M. Shah
Journal: Proc. Amer. Math. Soc. 93 (1985), 433-439
MSC: Primary 42A55; Secondary 42A28
DOI: https://doi.org/10.1090/S0002-9939-1985-0773997-X
MathSciNet review: 773997
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Abstract: P. B. Kennedy [3] studied lacunary Fourier series whose generating functions are of bounded variation on a subinterval $ I$ of $ [ - \pi ,\pi ]$ and satisfy a Lispschitz condition of order $ \alpha $ on $ I$. We show that the conclusion of one of his theorems on the absolute convergence of Fourier series remains valid when the function is merely of bounded $ r$th variation in $ I$ and belongs to a class $ \operatorname{Lip}(\alpha ,p)$ in $ I$. Our results also generalize three theorems of S. M. Mazhar [4].


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DOI: https://doi.org/10.1090/S0002-9939-1985-0773997-X
Keywords: Lacunary Fourier series, absolute convergence, Lipschitz condition, bounded $ r$th variation
Article copyright: © Copyright 1985 American Mathematical Society