On a distinctive property of Fourier bases associated with N- Bernoulli Convolutions

2025/05/31 by Zi-Chao Chi, Chi, Zi-Chao, Xing‐Gang He +3 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2506.00364

openalex publication_date 2025/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A distinctive problem of harmonic analysis on \R with respect to a Borel probability measure μ is identifying all t∈\R such that both \e^-2πiλx: λ∈Λ\\quadand \e^-2πiλx: λ∈ tΛ\ form orthonormal bases of the space L²(μ). Currently, this phenomenon has been observed only in certain singular measures. It is deeply connected to the convergence of Mock Fourier series with respect to the aforementioned bases. In this paper, we apply classical number theory to solve the general conjecture and basic problems in this field within the setting of N-Bernoulli convolutions, which extend almost all known results and give some new ones.

Citations

• Scaling of spectra of Cantor-type measures and some number theoretic considerations
• Number theory problems from the harmonic analysis of a fractal
• Spectra of Cantor measures

Cited by

• On infinite scalings of the canonical spectrum for self-similar spectral measures

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