Spectrality of random convolutions generated by finitely many Hadamard triples (2203.11619v3)

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 $\big{ e_\lambda(x) = e^{2\pi i \lambda x}: \lambda \in \Lambda_{\omega,{n_k}} \big}$ forms an orthonormal basis for $L^2(\mu_{\omega,{ n_k}})$.