A CLASS OF SPECTRAL MORAN MEASURES ON ℝ

Let [Formula: see text] be an arithmetic digit set for each [Formula: see text], where [Formula: see text], and let [Formula: see text] be a sequence of integers larger than 1. In this paper, we prove that the Moran measure [Formula: see text] generated by infinite convolution of finite atomic measures [Formula: see text] is a spectral measure if [Formula: see text] and [Formula: see text].

Spectrality of a Class of Moran Measures

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

EXPONENTIAL ORTHONORMAL BASES OF CANTOR–MORAN MEASURES

Let [Formula: see text] be a Cantor–Moran measure given by the infinite convolution of finite measures with equal probability [Formula: see text] where [Formula: see text] and [Formula: see text] for [Formula: see text] In this paper, we present a complete characterization for maximal orthogonal sets of exponentials of [Formula: see text] in terms of maximal mappings. As its application, we give a sufficient condition for a maximal orthogonal set to be a basis.

SPECTRAL PROPERTY OF CERTAIN MORAN MEASURES WITH THREE-ELEMENT DIGIT SETS

Let [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text], satisfy [Formula: see text]. Let [Formula: see text] be the infinite convolution of probability measures with finite support and equal distribution. In this paper, we show that if [Formula: see text], then there exists a discrete set [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text].