arXiv subjects

Wen-Hui Ai

Publications and source records attributed to Wen-Hui Ai.

5 recordsLinked to original sources

arXiv2026

Spectral structure and eigenmatrices of self-affine measures with $p$-digits in $ \bf{\mathbb{R}^{n}} $

For a prime number $p > 2$, let $\bm{0} \in D \subset \mathbb{Z}^n$ be a $p$-element digit set satisfying $ \mathcal{Z}(\widehatδ_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some $ \bm{a} \in \{ (i_1, \dots, i_n)^t : i_k \in [1, p-1] \cap \mathbb{Z}, 1\leq k\leq n \} $, where $\mathcal{Z}(\widehatδ_D)$ is the zero set of the Fourier transform of $δ_D$. Let $Q$ be an integer expansive diagonal matrix in $\mathbb{R}^{n}$, the self-affine measure $μ_{Q,D}$ is defined by \[ μ_{Q,D}(\cdot) = \frac{1}{\#D} \sum_{d \in D} μ_{Q,D}(Q(\cdot) - d). \] In this paper, we first provide sufficient condition for a maximal orthogonal family $E_Λ=\{ e^{-2πi \langle λ, x \rangle} : λ\in Λ\subset\mathbb{R}^{n}\}$ to be an orthogonal basis of $L^2(μ_{Q,D})$ when $Q = p\operatorname{diag}[q, \dots, q]$ with $q\geq 1$. Then we obtain necessary and sufficient conditions for the integer matrix $R$ such that $E_Λ$ and $E_{RΛ}$ are both orthogonal basis of $L^{2}(μ_{Q,D})$. Furthermore, for $Q = p\operatorname{diag}[l_1, \dots, l_n]$ with $|l_1 \dots l_n| > 1$, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of $μ_{Q,D}$.

math.FA↗

arXiv2025

Spectrality and non-spectrality of a class of Moran measures with three-element digits

A Borel probability measure $ μ$ with compact support on $ \mathbb{R}^n $ is called spectral measure if there exists a discrete set $ Λ\subset \mathbb{R}^n $ such that $ E_Λ:= \{e^{2πi \langle λ, x \rangle}: λ\in Λ\} $ forms an orthonormal basis of $ L^2(μ) $. In this paper, we study the spectrality and non-spectrality of a class of Moran measures with three-element digits on $ \mathbb{R} $. Let $p_n\in 3\mathbb Z\setminus\{0\}$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\{a_n,b_n\}=\{-1,1\}\pmod 3$. It is know that the infinite convolution of uniformly discrete probability measures $$μ_{\{p_n\},\{\mathcal D_n\}}:=δ_{p_1^{-1}\{0,a_1,b_1\}}\astδ_{(p_1p_2)^{-1}\{0,a_2,b_2\}}\ast\cdots $$ is a Moran measure with compact support if and only if \begin{align*} \sum_{n=1}^{\infty}|p_{1}p_{2}\cdots p_n|^{-1}d_n<\infty,\quad \mbox{where}\;d_n=\max\{0,|a_n|, |b_n|\}. \end{align*} Without the condition $\sup_{n\geq 1}\{\frac{|a_n|+|b_n|}{|p_n|}\}<\infty$, we give two sufficient conditions under which that $μ_{\{p_n\},\{\mathcal D_n\}}$ is a spectral measure. If $p_n=p>2$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\gcd(a_n,b_n)=1$, we also find an useful condition to guarantee that $μ_{p,\{\mathcal D_n\}}$ is not a spectral measure. Our results extend some known theorems in An et al. [JFA, 2019] and Lu et al. [JFAA, 2022].

math.FA↗

arXiv2025

On the Spectral Properties of a Class of Planar Sierpinski Self-Affine Measures

We investigate the spectral properties of a class of Sierpinski-type self-affine measures defined by \[ μ_{M,D}(\cdot) = p^{-1} \sum_{d \in D} μ_{M,D}(M(\cdot) - d), \] where $ p $ is a prime number, $ M = \begin{bmatrix} ρ_1^{-1} & c 0 & ρ_2^{-1} \end{bmatrix} $ is a real upper triangular expanding matrix, and $ D = \{d_0, d_1, \cdots, d_{p-1}\} \subset \mathbb{Z}^2 $ satisfying $ \mathcal{Z}(\widehatδ_{D}) = \cup_{j=1}^{p-1} \left( \frac{j \bm{a}}{p} + \mathbb{Z}^2 \right) $ for some $ \bm{a} \in \mathcal{E}_{p}= \{ (i_1, i_2)^* : i_1, i_2 \in [1, p-1] \cap \mathbb{Z} \} $, where $ \mathcal{Z}(\widehatδ_{D}) $ denotes the set of zeros of $ \widehatδ_{D} $ with $ δ_{D} = \frac{1}{\# D} \sum_{d \in D} δ_d $. When $ρ_1 = ρ_2$, we derive necessary and sufficient conditions for $μ_{M,D}$ to both: $(i)$ possess an infinite orthogonal set of exponential functions, and $(ii)$ be a spectral measure. When no infinite orthogonal exponential system exists in $L^{2}(μ_{M,D})$, we quantify the maximum number of orthogonal exponentials and provide precise estimates. For $ρ_1 \neq ρ_2$, with restricted digit sets $D$, we obtain a necessary and sufficient condition for $μ_{M,D}$ to be a spectral measure.

math.FA↗

arXiv2022

The divergence of Mock Fourier series for spectral measures

In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent on non-zero set. In particularly, there exists an example of the quarter Cantor measure whose Mock Fourier sums is not almost everywhere convergent.

math.FA↗

arXiv2022

Non-spectrality of Moran measures with consecutive digits

Let $ρ=(\frac{p}{q})^{\frac{1}{r}}<1$ for some $p,q,r\in\mathbb{N}$ with $(p,q)=1$ and $\mathcal{D}_{n}=\{0,1,\cdot\cdot\cdot,N_{n}-1\}$, where $N_{n}$ is prime for all $n\in\mathbb{N}$, and denote $M=\sup\{N_{n}:n=1,2,3,\ldots\}<\infty$. The associated Borel probability measure $$μ_{ρ,\{\mathcal{D}_{n}\}}=δ_{ρ\mathcal{D}_{1}}*δ_{ρ^{2}\mathcal{D}_{2}}*δ_{ρ^{3}\mathcal{D}_{3}}*\cdots$$ is called a Moran measure. Recently, Deng and Li proved that $μ_{ρ,\{\mathcal{D}_{n}\}}$ is a spectral measure if and only if $\frac{1}{N_{n}ρ}$ is an integer for all $n\geq 2$. In this paper, we prove that if $L^{2}(μ_{ρ, \{\mathcal{D}_{n}\}})$ contains an infinite orthogonal exponential set, then there exist infinite positive integers $n_{l}$ such that $(q,N_{n_{l}})>1$. Contrastly, if $(q,N_{n})=1$ and $(p,N_{n})=1$ for all $n\in\mathbb{N}$, then there are at most $M$ mutually orthogonal exponential functions in $L^{2}(μ_{ρ, \{\mathcal{D}_{n}\}})$ and $M$ is the best possible. If $(q,N_{n})=1$ and $(p,N_{n})>1$ for all $n\in\mathbb{N}$, then there are any number of orthogonal exponential functions in $L^{2}(μ_{ρ, \{\mathcal{D}_{n}\}})$.

math.FA↗