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On UC-multipliers for multiple trigonometric systems

Grigori A. Karagulyan

TL;DR

The paper studies which $w(n)$ can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. It develops a probabilistic equivalence framework using measure-preserving maps to relate one-dimensional and multidimensional systems. It shows that any such multiplier must satisfy $\log n \lesssim w(n) \lesssim \log^2 n$, and proves an equivalence principle under the growth condition $w(n^2) \le C w(n)$, effectively reducing the multidimensional problem to the 1D case. This enables extension of known 1D estimates to higher dimensions and yields parallel results for strong SRC multipliers.

Abstract

We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim w(n)\lesssim\log^2 n$. Our main result establishes a general equivalence principle between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions.

On UC-multipliers for multiple trigonometric systems

TL;DR

The paper studies which can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. It develops a probabilistic equivalence framework using measure-preserving maps to relate one-dimensional and multidimensional systems. It shows that any such multiplier must satisfy , and proves an equivalence principle under the growth condition , effectively reducing the multidimensional problem to the 1D case. This enables extension of known 1D estimates to higher dimensions and yields parallel results for strong SRC multipliers.

Abstract

We investigate the class of sequences that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds . Our main result establishes a general equivalence principle between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions.
Paper Structure (5 sections, 10 theorems, 51 equations)

This paper contains 5 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Weyl multipliers
  3. Probabilistically equivalence and measure-preserving maps
  4. A decomposition for discrete trigonometric systems
  5. Auxiliary lemmas and the proof of Theorem \ref{['T1']}

Key Result

Theorem A

The sequence $\log n$ is an RC-multiplier for the Haar system.

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Theorem A: Ul yanov, Uly4
  • Theorem B: Ul yanov, Uly4
  • Definition 1.3
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Corollary 1.1
  • Definition 2.1
  • ...and 15 more