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October 8, 2025Open Access

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

Key Points

Orthonormal bases of the space L2(mu) were identified for Fourier bases related to tLambda, notably in Bernoulli convolutions.
Key findings showed that certain singular measures permit the phenomenon where both bases are orthonormal, highlighting their unique properties.
Classical number theory approaches were utilized to explore problems within harmonic analysis, providing new insights and extensions of known results.
Implications of the findings include a deeper understanding of convergence related to Mock Fourier series, within the context of Bernoulli convolutions.

Abstract

A distinctive problem of harmonic analysis on with respect to a Borel probability measure is identifying all t such that both \\e^{-2 i x: \} \e^{-2 i x: t\}\ form orthonormal bases of the space L² (). 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.

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