Spectrality of random convolutions generated by finitely many Hadamard triples
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914631180615680 |
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| author | Li, Wenxia Miao, Jun Jie Wang, Zhiqiang |
| author_facet | Li, Wenxia Miao, Jun Jie Wang, Zhiqiang |
| contents | Let $\{(N_j, B_j, L_j): 1 \le j \le m\}$ be finitely many Hadamard triples in $\mathbb{R}$. Given a sequence of positive integers $\{n_k\}_{k=1}^\infty$ and $ω=(ω_k)_{k=1}^\infty \in \{1,2,\cdots, m\}^\mathbb{N}$, let $μ_{ω,\{n_k\}}$ be the infinite convolution given by $$μ_{ω,\{n_k\}} = δ_{N_{ω_1}^{-n_1} B_{ω_1}} * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} B_{ω_2}} * \cdots * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} \cdots N_{ω_k}^{-n_k} B_{ω_k} }* \cdots. $$ In order to study the spectrality of $μ_{ω,\{ n_k\}}$, we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if $\mathrm{gcd}(B_j - B_j)=1$ for $1 \le j \le m$, then all infinite convolutions $μ_{ω,\{n_k\}}$ are spectral measures. This implies that we may find a subset $Λ_{ω,\{n_k\}}\subseteq \mathbb{R}$ such that $\big\{ e_λ(x) = e^{2πi λx}: λ\in Λ_{ω,\{n_k\}} \big\}$ forms an orthonormal basis for $L^2(μ_{ω,\{ n_k\}})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_11619 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spectrality of random convolutions generated by finitely many Hadamard triples Li, Wenxia Miao, Jun Jie Wang, Zhiqiang Classical Analysis and ODEs Functional Analysis 28A80, 42C30 Let $\{(N_j, B_j, L_j): 1 \le j \le m\}$ be finitely many Hadamard triples in $\mathbb{R}$. Given a sequence of positive integers $\{n_k\}_{k=1}^\infty$ and $ω=(ω_k)_{k=1}^\infty \in \{1,2,\cdots, m\}^\mathbb{N}$, let $μ_{ω,\{n_k\}}$ be the infinite convolution given by $$μ_{ω,\{n_k\}} = δ_{N_{ω_1}^{-n_1} B_{ω_1}} * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} B_{ω_2}} * \cdots * δ_{N_{ω_1}^{-n_1} N_{ω_2}^{-n_2} \cdots N_{ω_k}^{-n_k} B_{ω_k} }* \cdots. $$ In order to study the spectrality of $μ_{ω,\{ n_k\}}$, we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if $\mathrm{gcd}(B_j - B_j)=1$ for $1 \le j \le m$, then all infinite convolutions $μ_{ω,\{n_k\}}$ are spectral measures. This implies that we may find a subset $Λ_{ω,\{n_k\}}\subseteq \mathbb{R}$ such that $\big\{ e_λ(x) = e^{2πi λx}: λ\in Λ_{ω,\{n_k\}} \big\}$ forms an orthonormal basis for $L^2(μ_{ω,\{ n_k\}})$. |
| title | Spectrality of random convolutions generated by finitely many Hadamard triples |
| topic | Classical Analysis and ODEs Functional Analysis 28A80, 42C30 |
| url | https://arxiv.org/abs/2203.11619 |