On the Coefficients of Multiple Walsh-Fourier Series with Small Gaps

For a Lebesgue integrable complex-valued function $f$ defined over the $m$-dimensional torus $\mathbb {I}^m:=[0,1)^m$, let $\hat f({\bf n})$ denote the multiple Walsh-Fourier coefficient of $f$, where ${\bf n}=\left(n^{(1)},\dots,n^{(m)}\right)\in (\mathbb {Z}^+)^m$, $\mathbb {Z^+}=\mathbb {N}\cup \{0\}$. The Riemann-Lebesgue lemma shows that $\hat f({\bf n})=o(1)$ as $|{\bf n}|\to \infty$ for any $f\in {\rm L}^1(\mathbb I^m)$. However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive result is due to Ghodadra Bhikha Lila for functions of bounded $p$-variation. We shall prove that this is just a matter only of local bounded $p$-variation for functions with multiple Walsh-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric For a Lebesgue integrable complex-valued function $f$ defined over the $m$-dimensional torus $\mathbb {I}^m:=[0,1)^m$, let $\hat f({\bf n})$ denote the multiple Walsh-Fourier coefficient of $f$, where ${\bf n}=\left(n^{(1)},\dots,n^{(m)}\right)\in (\mathbb {Z}^+)^m$, $\mathbb {Z^+}=\mathbb {N}\cup \{0\}$. The Riemann-Lebesgue lemma shows that $\hat f({\bf n})=o(1)$ as $|{\bf n}|\to \infty$ for any $f\in {\rm L}^1(\mathbb I^m)$. However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive result is due to Ghodadra Bhikha Lila for functions of bounded $p$-variation. We shall prove that this is just a matter only of local bounded $p$-variation for functions with multiple Walsh-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series due to J.R. Patadia and R.G. Vyas, illustrate the interconnection between `localness\u27 of the hypothesis and `type of lacunarity\u27 and allow us to interpolate the results