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

Anna Kazakova

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.

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

Abstract

We study problems on uniqueness sets (-sets) for multiple Walsh series converging over cubes and the properties of the coefficients of such series. New broad classes of -sets are constructed. In particular, it is proved that hyperplanes parallel to the coordinate ones are -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.
Paper Structure (11 sections, 17 theorems, 136 equations, 5 figures)

This paper contains 11 sections, 17 theorems, 136 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation, main definitions, and auxiliary facts
  3. Notation
  4. Main definitions
  5. Auxiliary statements
  6. Main results
  7. Dyadic planes on the group $\mathbb{G}^d$
  8. On the connection between quasimeasures on the group $\mathbb{G}^d$ and Walsh series
  9. On $U$-sets for convergence over cubes
  10. A Cantor–Lebesgue type theorem and a counterexample
  11. List of illustrations

Key Result

Lemma 2.1

Let $M_1, M_2 \in \mathbb{N}$, such that Then for the cubic partial sums $S_N$ of series (1.1) and for all $\mathbf{g} \in \mathbb{G}^d$, the following equality holds

Figures (5)

  • Figure 1: Layers of $M$-sets
  • Figure 2: Layers of Dirichlet-type sets
  • Figure 3: Dyadic planes of the form $D_1^{x}$
  • Figure 4: Dyadic planes of the form $Q_{1, \mathbf{q}}^{x}$ при $q_1 = 0$ и $q_2 =1$
  • Figure 5: Dyadic plane of the form $P_2^{\mathbf{x}}, \quad \mathbf{x} \in \mathbb{G}^2$

Theorems & Definitions (40)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Lemma 2.3
  • Proof 3
  • Lemma 2.4
  • Proof 4
  • Remark 1
  • Remark 2
  • ...and 30 more