On the coefficients of Vilenkin-Fourier series with small gaps

Abstract

The Riemann-Lebesgue lemma shows that the Vilenkin-Fourier coefficient f (n) is of o(1) as n → ∞ for any integrable function f on Vilenkin groups. However, it is known that the Vilenkin-Fourier coefficients of integrable functions can tend to zero as slowly as we wish. The definitive result is due to B. L. Ghodadra for functions of certain classes of generalized bounded fluctuations. 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.

On β-absolute convergence of Vilenkin-Fourier series with small gaps

“…Further, in [6], we have proved an analogue of the Wiener-Ingham inequality (see [6,Theorem 2]) and as its applications, extended the analogues on a Vilenkin group G of the well-known results of Bernstěin, Zygmund, Szász, and Stečhkin concerning the absolute convergence of Fourier series on G obtained by Vilenkin and Rubinstěin [22], Onneweer [9], and Quek and Yap ( [14,15]) for the lacunary Fourier series on G. 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. As in the case of trigonometric Fourier series (see [12]) and of our earlier results for Vilenkin-Fourier series (see [5,6]), here also we give an interconnection between the 'type of lacunarity' in Vilenkin-Fourier series and the localness of the hypothesis to be satisfied by the generic functions, which allow us to interpolate results concerning β-absolute convergence of lacunary and non-lacunary Vilenkin-Fourier series.…”