A note on the absolute convergence of lacunary Fourier series

Patel, N. V.; Shah, V. M.

doi:10.1090/S0002-9939-1985-0773997-X

A note on the absolute convergence of lacunary Fourier series
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by N. V. Patel and V. M. Shah

Proc. Amer. Math. Soc. 93 (1985), 433-439

DOI: https://doi.org/10.1090/S0002-9939-1985-0773997-X

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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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Trigonometrical series