论文标题
由有限的Hadamard三元组产生的随机卷积的光谱
Spectrality of random convolutions generated by finitely many Hadamard triples
论文作者
论文摘要
令$ \ {（n_j，b_j，l_j）：1 \ le j \ le m \} $在$ \ mathbb {r} $中有限许多Hadamard Triples。给定一系列正整数$ \ {n_k \} _ {k = 1}^\ infty $和$ω=（ω__k）_ {k = 1}^\ in \ in \ in \ in \ in \ in \ in \ {1,2，\ cdots，\ cdots，m \} $μ_{ω，\ {n_k \}} $是由$$μ_{ω，\ {n_k \}} =δ__{n_ {ω_1}^n_1}^ - n_1}^{ - n_ {ω_2}^{ - n_2} b_ {ω__2}} * \ cdots *δ__{n_ {n_ {ω_1}^{ - n_1} n_1} n_ {ω_2}} \ cdots。 $$为了研究$μ_{ω，\ {n_k \}} $的光谱，我们首先显示了在等积极性条件下Hadamard Triples产生的一般无限卷积的光谱。然后，通过使用傅里叶变换的整体周期性零集，我们表明，如果$ \ mathrm {gcd}（b_j -b_j）= 1 $ for $ 1 \ le j \ le j \ le m $，则所有无限卷积$μ__{ω，\ {n_k \ \ \ \}} $是光谱尺寸。这意味着我们可能会找到一个子集$λ_{ω，\ {n_k \}} \ subseteq \ subseteq \ mathbb {r} $，这样，$ \ big \ big \ {e_λ（x）= e^e^{2πiλx} $ l^2（μ_{ω，\ {n_k \}}）$的正交基础。
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 $L^2(μ_{ω,\{ n_k\}})$.
