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

A. D. Kazakova, M. G. Plotnikov

TL;DR

This work extends Menšhov-type null-series theory to the $d$-dimensional Walsh system by constructing explicit $M$-sets using nested graphs of Walsh functions within dyadic cubes. A sequence $(m_s)$ defines sets $F_s$ whose intersection $F$ yields an $M$-set under convergence over rectangles, cubes, and iterated orders, with a corresponding null-series realizing $F$. Nonempty portions of $F$ remain $M$-sets, and symmetry via permutations produces additional $M$-sets and, under suitable conditions, $U$-sets, all described through a quasi-measure framework linking Walsh series and measures. The results generalize multidimensional uniqueness/nonuniqueness phenomena and provide a detailed geometric and analytic structure for $M$/$U$-set classifications in multidimensional Walsh contexts.

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

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

TL;DR

This work extends Menšhov-type null-series theory to the -dimensional Walsh system by constructing explicit -sets using nested graphs of Walsh functions within dyadic cubes. A sequence defines sets whose intersection yields an -set under convergence over rectangles, cubes, and iterated orders, with a corresponding null-series realizing . Nonempty portions of remain -sets, and symmetry via permutations produces additional -sets and, under suitable conditions, -sets, all described through a quasi-measure framework linking Walsh series and measures. The results generalize multidimensional uniqueness/nonuniqueness phenomena and provide a detailed geometric and analytic structure for /-set classifications in multidimensional Walsh contexts.

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

Paper Structure

This paper contains 7 sections, 17 theorems, 99 equations.

Table of Contents

  1. Preliminaries
  2. Notations
  3. Basic Definitions and Preliminary Facts
  4. Auxiliary results
  5. Construction of $M$-set for multiple Walsh series and null-series
  6. Auxliary lemmas
  7. Main results

Key Result

Proposition 1.1

Suppose the indices of all nonzero terms of the $d$-dim numerical series lie in the set $\{ \mathbf{0} \} \bigsqcup \bigsqcup\limits_{k \in \mathbb{N}_0} B_k$, where, recall that, $B_k := \{ \mathbf{n} \in \mathbf{N} ^d \colon 2^k \mathbf{1} \le \mathbf{n} < 2^{k + 1} \mathbf{1} \}$, and the series itself converges to the sum $S$ over rectangles. Then it converges to

Theorems & Definitions (33)

  • Proposition 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • ...and 23 more