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- W4225501239 abstract "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 $omega=(omega_k)_{k=1}^infty in {1,2,cdots, m}^mathbb{N}$, let $mu_{omega,{n_k}}$ be the infinite convolution given by $$mu_{omega,{n_k}} = delta_{N_{omega_1}^{-n_1} B_{omega_1}} * delta_{N_{omega_1}^{-n_1} N_{omega_2}^{-n_2} B_{omega_2}} * cdots * delta_{N_{omega_1}^{-n_1} N_{omega_2}^{-n_2} cdots N_{omega_k}^{-n_k} B_{omega_k} }* cdots. $$ In order to study the spectrality of $mu_{omega,{ n_k}}$, 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 $mu_{omega,{n_k}}$ are spectral measures. This implies that we may find a subset $Lambda_{omega,{n_k}}subseteq mathbb{R}$ such that $big{ e_lambda(x) = e^{2pi i lambda x}: lambda in Lambda_{omega,{n_k}} big}$ forms an orthonormal basis for $L^2(mu_{omega,{ n_k}})$." @default.
- W4225501239 created "2022-05-05" @default.
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- W4225501239 date "2022-03-22" @default.
- W4225501239 modified "2023-10-12" @default.
- W4225501239 title "Spectrality of random convolutions generated by finitely many Hadamard triples" @default.
- W4225501239 doi "https://doi.org/10.48550/arxiv.2203.11619" @default.
- W4225501239 hasPublicationYear "2022" @default.
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