Uncertainty product for Vilenkin groups

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products [Formula: see text] and [Formula: see text] that are similar to the Heisenberg uncertainty product. [Formula: see text] and [Formula: see text] differ from each other by the metric used for the Vilenkin group [Formula: see text]. We discuss analogs of a quantitative uncertainty principle. Representations for [Formula: see text] and [Formula: see text] in terms of Walsh and Haar basis are given.

Subsequences of triangular partial sums of double fourier series on unbounded Vilenkin groups

In 1987 Harris proved-among others-that for each 1 ? p < 2 there exists a two-dimensional function f ? Lp such that its triangular partial sums S?2A f of Walsh-Fourier series does not converge almost everywhere. In this paper we prove that subsequences of triangular partial sums S?nAMAf,nA ? {1,2, ...,mA-1} on unbounded Vilenkin groups converge almost everywhere to f for each function f ? L2.