On multiple null-series in the Walsh system, M- and U- sets

TL;DR

This paper constructs specific M-sets and null-series within the d-dimensional Walsh system, analyzing their convergence properties and how to modify them into U-sets, advancing understanding of convergence behaviors in Walsh systems.

Contribution

It introduces a new construction of M-sets and null-series in the Walsh system considering various convergence modes, and explores their modification into U-sets.

Findings

Constructed M-sets and null-series for Walsh systems over different convergence modes.

Analyzed the rate of convergence of coefficients in zero-series realizing M-sets.

Provided a method to modify M-sets into U-sets.

Abstract

A family of M-sets and null-series for the d-dimensional Walsh system is constructed if we consider convergence over rectangles, cubes, or iterated convergence. Non-empty portions of the constructed M-sets are also M-sets. The question of the rate of convergence to zero of the coefficients of zero-series that realize the constructed M-sets is studied, and it is shown how to modify the construction of the latter to turn them into U-sets