Almost everywhere convergence of Mock Fourier series for the standard middle-fourth Cantor measure with respect to its canonical spectrum

2026/07/30 by Min-Wei Tang, Zhi-Yi Wu
Mathematics · #math.CA

paper · pdf

16 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system \(x-1)/(4),(x+1)/(4)\ with equal weights, denoted by μ_1/4, is a spectral measure with a spectrum Λ₄, called the canonical spectrum,Λ₄ := \set ∑_j=0^m-1ε_j4^j: m≥ 1, ε_j∈\0,1\ . For f∈ L¹(μ_1/4), let S_n f be the n-th partial sum of its Mock Fourier series with respect to Λ₄. We prove that the associated maximal operator S^∗f=sup_n|S_n f| is of weak type (1,1). Consequently, S_n f→ f μ_1/4-almost everywhere on \supp(μ_1/4). This solves a long-standing open problem of Strichartz \cite[p.~341]Str06.

Related

• The divergence of Mock Fourier series for spectral measures
• The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin
• Spectra of Cantor measures
• Divergence of mock and scrambled Fourier series
• Additive spectra of the 1/4 Cantor measure