论文标题
无限卷积的收敛性和光谱较弱
Weak Convergence and Spectrality of Infinite Convolutions
论文作者
论文摘要
令$ \ {a_k \} _ {k = 1}^\ infty $是$ \ m athbb {r}^d $满足所有$ \＃a_k \ ge 2 $的有限子集的序列。在本文中，我们首先为存在无限卷积的$$ n n n n um \ frac {1} {\＃a} \ sum_ {a \ in}Δ_a$。然后，我们研究Hadamard Triples在$ \ Mathbb {R} $中产生的一类无限卷积的光谱，并在没有紧凑的支持的情况下构建了一类奇异光谱测量。最后，我们证明了这种措施很丰富，其支持的维度具有中间值。
Let $\{ A_k\}_{k=1}^\infty$ be a sequence of finite subsets of $\mathbb{R}^d$ satisfying that $\# A_k \ge 2$ for all integers $k \ge 1$. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution $$ν=δ_{A_1}*δ_{A_2} * \cdots *δ_{A_n}*\cdots, $$ where all sets $A_k \subseteq \mathbb{R}_+^d$ and $δ_A = \frac{1}{\# A} \sum_{a \in A} δ_a$. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in $\mathbb{R}$ and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.
