Canadian Mathematical Bulletin ( IF 0.6 ) Pub Date : 2020-01-17 , DOI: 10.4153/s000843951900047x Ming-Liang Chen , Jing-Cheng Liu , Juan Su , Xiang-Yang Wang
Let $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$ , and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$ , where $p_{n}$ , $q_{n}$ , $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$ . If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$ , then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$ .
中文翻译:
一类Moran测度的谱
假设 $ \ {M_ {n} \} _ {n = 1} ^ {\ infty} $ 是具有 $ M_ {n} = \ operatorname {diag}(p_ {n},q_ {n} )$ ,并让 $ \ {{\ mathcal {D}} _ {n} \} _ {n = 1} ^ {\ infty} $ 是具有 $ {\ mathcal {D}} _ { n} = \ {(0,0)^ {t},(a_ {n},0)^ {t},(0,b_ {n})^ {t},\ pm(a_ {n},b_ {n})^ {t} \} $ ,其中 $ p_ {n} $ , $ q_ {n} $ , $ a_ {n} $ 和 $ b_ {n} $ 是所有 $ n \ geqslant 1的正整数$ 。如果 $ \ sup _ {n \ geqslant 1} \ {\ frac {a_ {n}} {p_ {n}},\ frac {b_ {n}} {q_ {n}} \} <\ infty $ ,则无限卷积 $ \ unicode [STIX] {x1D707} _ {\ {M_ {n} \},\ {{\ mathcal {D}} _ {n} \}} = \ unicode [STIX] {x1D6FF} _ {M_ {1 } ^ {-1} {\ mathcal {D}} _ {1}} \ ast \ unicode [STIX] {x1D6FF} _ {(M_ {1} M_ {2})^ {-1} {\ mathcal {D }} _ {2}} \ ast \ cdots \,$ 是Borel概率度量(Cantor–Dust–Moran度量)。在本文中,我们研究了何时存在离散集 $ \ unicode [STIX] {x1D6EC} $ ,使得 $ \ {e ^ {2 \ unicode [STIX] {x1D70B} i \ langle \ unicode [STIX] {x1D706} ,x \ rangle}:\ unicode [STIX] {x1D706} \}中的\ unicode [STIX] {x1D706} \ $ 是 $ L ^ {2}(\ unicode [STIX] {x1D707} _ {\ {M_ {n} \},\ {{\ mathcal {D}} _ {n} \}})$ 。