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

TL;DR

This paper investigates the unique properties of Fourier bases associated with N-Bernoulli convolutions, solving a general conjecture in harmonic analysis and extending known results through number theory techniques.

Contribution

It applies classical number theory to resolve a broad conjecture about Fourier bases in N-Bernoulli convolutions, advancing the understanding of harmonic analysis on fractal measures.

Findings

Identifies all t for which scaled Fourier bases form orthonormal bases in L^2(μ)

Extends known results on Fourier bases for singular measures

Provides new insights into convergence of Fourier series on fractal measures

Abstract

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi i\lambda x}: \lambda\in t\Lambda\right\}\] form orthonormal bases of the space $L^2(\mu)$. 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.