## ON THE COEFFICIENTS OF MULTIPLE WALSH-FOURIER SERIES WITH SMALL GAPS

JMS
Vol. 23 (2016), No. 4, Page 727–740.

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' of the hypothesis and type of lacunarity' and allow us to interpolate the results.