arXiv subjects

Ming-Liang Chen

Publications and source records attributed to Ming-Liang Chen.

3 recordsLinked to original sources

arXiv2022

Fourier bases of a class of planar self-affine measures

Let $μ_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\in M_2(\mathbb{Z})$ and a non-collinear integer digit set $D=\left\{\begin{pmatrix} 0\\0\end{pmatrix},\begin{pmatrix} α_{1}\\ α_{2} \end{pmatrix}, \begin{pmatrix} β_{1}\\ β_{2} \end{pmatrix}, \begin{pmatrix} -α_{1}-β_{1}\\ -α_{2}-β_{2} \end{pmatrix}\right\}$. In this paper, we show that $μ_{M,D}$ is a spectral measure if and only if there exists a matrix $Q\in M_2(\mathbb{R})$ such that $(\tilde{M},\tilde{D})$ is admissible, where $\tilde{M}=QMQ^{-1}$ and $\tilde{D}=QD$. In particular, when $α_1β_2-α_2β_1\notin 2\Bbb Z$, $μ_{M,D}$ is a spectral measure if and only if $M\in M_2(2\mathbb{Z})$.

math.FA↗

arXiv2020

Spectrality of generalized Sierpinski-type self-affine measures

For an expanding integer matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $D=\{(0,0)^t,(α_1,α_2)^t,(β_1,β_2)^t\}$ with $α_1β_2-α_2β_1\neq0$, let $μ_{M,D}$ be the Sierpinski-type self-affine measure defined by $μ_{M,D}(\cdot)=\frac{1}{3}\sum_{d\in D}μ_{M,D}(M(\cdot)-d)$. In [5.36], the authors separately investigated the spectral property of the measure $μ_{M,D}$ in the case of $\det(M)\notin 3\mathbb{Z}$ or $α_1β_2-α_2β_1\notin 3\mathbb{Z}$. In this paper, we consider the remaining case where $\det(M)\in 3\mathbb{Z}$ and $α_1β_2-α_2β_1\in 3\mathbb{Z}$, and give the necessary and sufficient conditions for $μ_{M,D}$ to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure $μ_{M,D}$.

math.CA↗

arXiv2018

The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets

In this paper, we consider the planar self-affine measures $μ_{M,D}$ generated by an expanding matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $ D=\left\{ {\left( {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} α_1\\ α_2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} β_1\\ β_2 \end{array}} \right)} \right\} $ with $α_1β_2-α_2β_1\neq0$. We show that if $\det(M)\notin 3\mathbb{Z}$, then the mutually orthogonal exponential functions in $L^2(μ_{M,D})$ is finite, and the exact maximal cardinality is given.

math.FA↗