On uniqueness sets and coefficients of multiple Walsh series converging over cubes

TL;DR

This paper investigates the properties of uniqueness sets for multiple Walsh series over cubes, introduces new classes of such sets including hyperplanes, and characterizes the behavior of their coefficients.

Contribution

It constructs new broad classes of $U$-sets, including hyperplanes, and describes the behavior of coefficients for converging Walsh series over cubes.

Findings

Hyperplanes parallel to coordinate axes are $U$-sets.

Characterization of index sets where coefficients grow arbitrarily large.

Description of index sets where coefficients tend to zero.

Abstract

We study problems on uniqueness sets ($U$-sets) for multiple Walsh series converging over cubes and the properties of the coefficients of such series. New broad classes of $U$-sets are constructed. In particular, it is proved that hyperplanes parallel to the coordinate ones are $U$-sets. For the coefficients of multiple Walsh series converging over cubes, both the index sets on which they can be made arbitrarily large and the index sets on which these coefficients tend to zero are described.