On the coefficients of Vilenkin-Fourier series with small gaps

Abstract

The Riemann-Lebesgue lemma shows that the Vilenkin-Fourier coefficient $\mathrm{f̂}\left(n\right)$ is of $o\left(1\right)$ as $n\to \infty$ 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.