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

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

Ghodadra, Bhikha Lila
ON THE COEFFICIENTS OF MULTIPLE WALSH-FOURIER SERIES WITH SMALL GAPS
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Abstract:
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.

Keywords: Multiple Walsh-Fourier coe cient, Function of bounded p-variation in several variables, order of magnitude, Lacunary Fourier series with small gaps.

Mathematics Subject Classification (2010): 42C10, 42B05, 26B30, 26D15.
Mathematical Reviews Number: MR3588260

Received: 2014-11-17