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On the absolute convergence of lacunary Fourier series


Author: J. R. Patadia
Journal: Proc. Amer. Math. Soc. 71 (1978), 19-25
MSC: Primary 42A44
MathSciNet review: 0493138
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Abstract: Let $ f \in L[ - \pi ,\pi ]$ be $ 2\pi $-periodic. Noble [6] posed the following problem: if the fulfillment of some property of a function f on the whole interval $ [ - \pi ,\pi ]$ implies certain conclusions concerning the Fourier series $ \sigma (f)$ of f, then what lacunae in $ \sigma (f)$ guarantees the same conclusions when the property is fulfilled only locally? Applying the more powerful methods of approach to this kind of problems, originally developed by Paley and Wiener [7], the absolute convergence of a certain lacunary Fourier series is studied when the function f satisfies some hypothesis in terms of either the modulus of continuity or the modulus of smoothness of order l considered only at a fixed point of $ [ - \pi ,\pi ]$. The results obtained here are a kind of generalization of the results due to Patadia [8].


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0493138-2
Keywords: Lacunary Fourier series, absolute convergence, modulus of continuity, Hadamard lacunarity condition
Article copyright: © Copyright 1978 American Mathematical Society