Spectrality of a Class of Moran Measures. Issue 2 (17th June 2020)

• Record Type: Journal Article
• Authors: Chen, Ming-Liang; Liu, Jing-Cheng; Su, Juan; Wang, Xiang-Yang
• Is Part Of: Canadian mathematical bulletin =. Volume 63:Issue 2(2020)
• Page Start: 366
• Page End: 381
• Publication Date: 2020-06-17
• Subjects: 28A80, -- 42C05, -- 46C05; Cantor–Dust–Moran measure, -- spectral measure, -- spectrum, -- infinite convolution; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S000843951900047X
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: 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 $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\}, \{{\mathcal{D}}_{n}\}})$ .