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For a Lebesgue integrable complex-valued function $f$ defined on ${ℝ}^{+}:=[0,\infty )$ let $\widehat{f}$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\widehat{f}\left(y\right)\to 0$ as $y\to \infty$. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1\left({ℝ}^{+}\right)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We...

We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\widehat{f}\left(n\right)$, $n\in \mathbb{N}$, of functions $f\in L^p\left(G\right)$ for some $1<p\le 2$. We obtain certain sufficient conditions for the finiteness of the series ${\sum }_{n=1}^{\infty }{a}_n{\left|\widehat{f}\left(n\right)\right|}^r$, where $\left\{a_n\right\}$ is a given sequence of positive real numbers satisfying a mild assumption and $0<r<2$. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative analogue...