On the absolute convergence of lacunary Fourier series

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].