On $\beta$-absolute Convergence of Vilenkin-Fourier Series with Small Gaps

Bhikha Lila Ghodadra

The study of absolute convergence of Fourier series is one of the most important problems of Fourier Analysis and the problem has been studied intensively by many researchers in the setting of circle group in particular and classical groups in general. Recently, in [Math. Inequal. Appl. { 17} (2) (2014), 749-760], we have studied the $\beta$-absolute convergence ($0<\beta\le 2$) of Vilenkin-Fourier series for the functions of various classes of functions of generalized bounded fluctuation and given sufficient conditions in terms of modulus of continuity. In this paper, we prove that this is a matter only of local fluctuation for functions with the Vilenkin-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series, illustrate the interconnection between `localness' of the hypothesis and type of lacunarity and allow us to interpolate the results.