On the absolute convergence of lacunary Fourier series
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 by J. R. Patadia PDF
 Proc. Amer. Math. Soc. 71 (1978), 1925 Request permission
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].References

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Additional Information
 © Copyright 1978 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 71 (1978), 1925
 MSC: Primary 42A44
 DOI: https://doi.org/10.1090/S00029939197804931382
 MathSciNet review: 0493138