Fourier Series with Small Gaps and Functions of Generalized Variations

Considering the lacunary Fourier series ∑∞− ∞ƒ̂(nk) exp(inkx) (n−k = −nk) on the circle group with small gaps (nk+1−nk) ≥ q ≥ 1, (1) we apply the Wiener-Ingham result for the finite trigonometric sums with small gaps and the technique developed earlier by us to study the sufficiency conditions in terms of various generalised bounded variations and the modulus/quadratic modulus of continuity (considered only locally) for the absolute convergence of such series. The precise and beautiful interconnection between the type of lacunarity in such series and the localness of the hypothesis to be satisfied by the generic function allows us to interpolate results concerning lacunary series and nonlacunary series (equality throughout in (1)). We make precise some earlier results and generalise several known results of this kind for such series.