Absolute convergence of the Fourier trigonometric series with gaps

In the present paper, the sufficient conditions are obtained for the generalized beta-absolute convergence (0 < beta < 2) of the Fourier trigonometric series with gaps for some classes of functions. In [8], analogous problems were considered for Fourier trigonometric series and sufficient conditions were established in terms of the delta-variation of a function; also, it was proved that these conditions are unimprovable in a certain sense. Our goal is to show that if a function f has a Fourier series with gaps, then the results obtained in [8] hold if the function f satisfies the derived conditions only on an arbitrarily small interval.