On null-series by double Walsh system
The author obtains 2-dimensional analogues of certain 1-dimensional results of \textit{G. G. Gevorkian} [Anal. Math. 14, No. 3, 219-251 (1988; Zbl 0713.42025)]. For example, he proves that if \(\varepsilon_{mn}\) satisfies certain growth conditions (which include a mild monotone property) and \(\sum \varepsilon^2_{mn} = \infty\), then there is a subset \(E\) of the unit square of two-dimensional Lebesgue measure zero such that given any double Walsh series, which converges to zero except possibly on countably many line segments parallel to the axes, whose coefficients satisfy \(b_{mn} = o(\varepsilon_{mn})\), it must be the case that \(b_{mn} = 0\) for all \(m\) and \(n\). He also shows this result is false if \(b_{mn} = o (\varepsilon_{ mn})\) is replaced by \(b_{mn} = O (\varepsilon_{mn})\). The proof is constructive and somewhat complicated. It relies on the uniqueness result of \textit{H. O. Movsisjan} [Izv. Akad. Nauk. Arm. SSR, Mat. 9, 40-61 (1974; Zbl 0294.42009)].
scientific article; zbMATH DE number 795431