Let f ∈ L [ − π , π ] f \in L[ - \pi ,\pi ] be 2 π 2\pi -periodic. Noble [6] posed the following problem: if the fulfillment of some property of a function f on the whole interval [ − π , π ] [ - \pi ,\pi ] implies certain conclusions concerning the Fourier series σ ( f ) \sigma (f) of f, then what lacunae in σ ( f ) \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 [ − π , π ] [ - \pi ,\pi ] . The results obtained here are a kind of generalization of the results due to Patadia [8].