Spectrality of random convolutions generated by finitely many Hadamard triples
2023; IOP Publishing; Volume: 37; Issue: 1 Linguagem: Inglês
10.1088/1361-6544/ad0d70
ISSN1361-6544
AutoresWenxia Li, Jun Jie Miao, Zhiqiang Wang,
Tópico(s)Advanced Algebra and Geometry
ResumoAbstract Let <?CDATA $\{(N_j, B_j, L_j): 1 \unicode{x2A7D} j \unicode{x2A7D} m\}$?> { ( N j , B j , L j ) : 1 ⩽ j ⩽ m } be finitely many Hadamard triples in <?CDATA $\mathbb{R}$?> R . Given a sequence of positive integers <?CDATA $\{n_k\}_{k = 1}^\infty$?> { n k } k = 1 ∞ and <?CDATA $\omega = (\omega_k)_{k = 1}^\infty \in \{1,2,\ldots, m\}^{\mathbb{N}}$?> ω = ( ω k ) k = 1 ∞ ∈ { 1 , 2 , … , m } N , let <?CDATA $\mu_{\omega,\{n_k\}}$?> μ ω , { n k } be the infinite convolution given by <?CDATA $\mu_{\omega,\left\{n_k\right\}} = \delta_{N_{\omega_1}^{-n_1} B_{\omega_1}} * \delta_{N_{\omega_1}^{-n_1} N_{\omega_2}^{-n_2} B_{\omega_2}} * \cdots * \delta_{N_{\omega_1}^{-n_1} N_{\omega_2}^{-n_2} \cdots N_{\omega_k}^{-n_k} B_{\omega_k} }* \cdots. $?> 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 ∗ ⋯ ∗ δ N ω 1 − n 1 N ω 2 − n 2 ⋯ N ω k − n k B ω k ∗ ⋯ . In order to study the spectrality of <?CDATA $\mu_{\omega,\{n_k\}}$?> 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 <?CDATA $\mathrm{gcd}(B_j - B_j) = 1$?> g c d ( B j − B j ) = 1 for <?CDATA $1 \unicode{x2A7D} j \unicode{x2A7D} m$?> 1 ⩽ j ⩽ m , then all infinite convolutions <?CDATA $\mu_{\omega,\{n_k\}}$?> μ ω , { n k } are spectral measures. This implies that we may find a subset <?CDATA $\Lambda_{\omega,\{n_k\}}\subseteq \mathbb{R}$?> Λ ω , { n k } ⊆ R such that <?CDATA $\big\{e_\lambda(x) = e^{2\pi i \lambda x}: \lambda \in \Lambda_{\omega,\{n_k\}} \big\}$?> { e λ ( x ) = e 2 π i λ x : λ ∈ Λ ω , { n k } } forms an orthonormal basis for <?CDATA $L^2(\mu_{\omega,\{n_k\}})$?> L 2 ( μ ω , { n k } ) .
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