On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912405732196352 |
|---|---|
| author | Chi, Zi-Chao He, Xing-Gang Wu, Zhi-Yi |
| author_facet | Chi, Zi-Chao He, Xing-Gang Wu, Zhi-Yi |
| contents | A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $μ$ is identifying all $t\in\R$ such that both \[\left\{e^{-2πiλx}: λ\inΛ\right\}\quad\text{and}\quad \left\{e^{-2πiλx}: λ\in tΛ\right\}\] form orthonormal bases of the space $L^2(μ)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions Chi, Zi-Chao He, Xing-Gang Wu, Zhi-Yi Classical Analysis and ODEs A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $μ$ is identifying all $t\in\R$ such that both \[\left\{e^{-2πiλx}: λ\inΛ\right\}\quad\text{and}\quad \left\{e^{-2πiλx}: λ\in tΛ\right\}\] form orthonormal bases of the space $L^2(μ)$. 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. |
| title | On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2506.00364 |