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The divergence of Mock Fourier series for spectral measures

Wu-Yi Pan, Wen-Hui Ai

TL;DR

This work analyzes divergence phenomena for Mock Fourier series on doubling spectral measures, establishing a sufficient condition—based on tail behavior of the Mock Dirichlet operator—that guarantees the existence of an integrable function with divergence on a μ-nonzero set. The authors develop a dyadic-cube framework and maximal operator arguments to translate tail estimates for discrete Dirac measures into L^1 divergence results, connecting convergence questions to kernel maximal functions. They then apply the criterion to self-affine spectral measures generated by Hadamard triples, deriving a concrete condition Δ(m_τ,b)>1 that implies divergence and yielding a corollary for the quarter Cantor measure with a specific spectrum (e.g., 17Λ). Overall, the paper advances understanding of convergence vs. divergence for Fourier-type expansions on fractal spectral measures and provides tools that bridge harmonic analysis, fractal geometry, and ergodic encodings.

Abstract

In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent on non-zero set. In particularly, there exists an example of the quarter Cantor measure whose Mock Fourier sums is not almost everywhere convergent.

The divergence of Mock Fourier series for spectral measures

TL;DR

This work analyzes divergence phenomena for Mock Fourier series on doubling spectral measures, establishing a sufficient condition—based on tail behavior of the Mock Dirichlet operator—that guarantees the existence of an integrable function with divergence on a μ-nonzero set. The authors develop a dyadic-cube framework and maximal operator arguments to translate tail estimates for discrete Dirac measures into L^1 divergence results, connecting convergence questions to kernel maximal functions. They then apply the criterion to self-affine spectral measures generated by Hadamard triples, deriving a concrete condition Δ(m_τ,b)>1 that implies divergence and yielding a corollary for the quarter Cantor measure with a specific spectrum (e.g., 17Λ). Overall, the paper advances understanding of convergence vs. divergence for Fourier-type expansions on fractal spectral measures and provides tools that bridge harmonic analysis, fractal geometry, and ergodic encodings.

Abstract

In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent on non-zero set. In particularly, there exists an example of the quarter Cantor measure whose Mock Fourier sums is not almost everywhere convergent.
Paper Structure (3 sections, 11 theorems, 76 equations)

This paper contains 3 sections, 11 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The proof of theorem \ref{['thm1.1']}
  3. Application to self-affine spectral measures

Key Result

Theorem 1.1

Assume $\mu$ is a doubling spectral measure. Let $(S_n,\Lambda_n)$ be the Mock Dirichlet summation operator and if then there exists an integrable function such that the Mock Fourier series diverges on $\mu$-non-zero set.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • Definition 3.1: Self-affine measure
  • Definition 3.2: Hadamard triple
  • ...and 6 more