arXiv subjects

Wei-Jie Wang

Publications and source records attributed to Wei-Jie Wang.

6 recordsLinked to original sources

arXiv2026

Nonexistence of Frame Measures for Separated Uniform Consecutive-Digit Bernoulli Convolutions

We investigate the existence problem of frame measures for uniform consecutive-digit Bernoulli convolutions under the separated scaling condition. Given parameters $N\geq 2$ and $0<ρ<1/N$, we prove that if $ρ^{-m}=B$ for some integers $m\geq 1$ and $B\geq 2$, with $N\nmid B$, then the associated $N$-Bernoulli convolution $μ_{ρ,N}$ admits no frame measure. The proof introduces a new cyclic mask-quotient obstruction adapted to the multi-character structure of the consecutive-digit framework.

math.FA↗

arXiv2026

Turán problem on the union of three intervals of equal length

This paper addresses the Turán extremal problem on the symmetric three-interval set $Ω_λ= (-1,1) \cup (λ-1,λ+1) \cup (-λ-1,-λ+1)$, $λ\geq 0$. We establish a discrete approximation principle relating the continuous Turán problem on $Ω_λ$ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters $λ$. We further analyze the dependence of the Turán constant on $λ$ and derive upper bounds for all $λ> 5$.

math.CA↗

arXiv2026

Spectral eigenvalue set of self-similar measures associated with product-form Hadamard triples

Previously, An \cite{AL01} showed that the self-similar measure $μ$ generated by a product-form Hadamard triple is a spectral measure. In this paper, we study its spectral eigenvalue problem. A set $A\subset\mathbb R$ is called a spectral eigenvalue set of $μ$ if there exists a spectrum $Λ$ of $μ$ such that $aΛ$ is a spectrum of $μ$ for every $a\in A$. We introduce the Product-form Hadamard multiplier set $\mathcal{T}_*$, and prove that for any $s\in [0,\frac{\log \#\mathcal{D}}{\log N}]$, the spectral eigensubspace $$V^{(s)}(μ_{N,\mathcal{D}},\mathcal{T}_*):=\{Λ:t Λ\text{ is a spectrum of }μ\text{ for all }t \in\mathcal{T}_* \text{ and } \dim_{Be}(Λ)=s\}$$ has the cardinality of the continuum. This result allows us to show that for the four-digit self-similar measures, a real number $t$ is a spectral eigenvalue if and only if $t \in \left\{\frac{u}{v}:u,v\in 2\mathbb{Z}+1\right\}$. And for any subset $S$ of $\mathbb{R}$ is a spectral eigenvalue set if and only if $S \subset t^{-1} (2\mathbb{Z}+1)$ for some $t\in 2\mathbb{Z}+1$.

math.FA↗

arXiv2026

A Walsh-Quotient Obstruction for Fourier Frames on Odd Reciprocal-Power Bernoulli Convolutions

We introduce a Walsh--quotient obstruction to study Fourier-frame existence for symmetric two-branch Bernoulli convolutions \[ μ_{ρ,d} =\ast_{j=1}^{\infty} \frac12\bigl(δ_{-dρ^{j}/2}+δ_{dρ^{j}/2}\bigr), \qquad 0<ρ<1,\quad d>0. \] Suppose that $0<ρ<\frac12$ and $ρ^{-m}=B$ for some integer $m\ge1$ and odd integer $B\ge3$. We prove that $L^2(μ_{ρ,d})$ admits no Fourier frame. For $m=1$, our argument proves the nonexistence of Fourier frames for odd-integer-base Cantor measures and hence resolves Strichartz's long-standing open problem for the middle-third Cantor measure. A contemporaneous independent proof of the case $m=1$ was obtained by Pont, Liehr and Taylor [arXiv:2607.08656v1]. For $m>1$, our theorem includes the non-integer reciprocal-power contraction ratios $ρ=B^{-1/m}$, which fall outside the classical integer-base Cantor-measure setting. Our proof is self-contained. It uses finite-coordinate Walsh packets to transform the frame inequalities into incompatible tangent-quotient estimates, while the identity $ρ^{-m}=B$ supplies the exact $m$-step scale relation leading to the contradiction.

math.FA↗

arXiv2026

Common Spectral Eigenvalue Spectra for Random Convolutions Generated by Hadamard Triples

Lu proved that the set $$\mathcal{T}:=\{t\in\mathbb{Z}\setminus\{0\}:(q,\mathcal{D},tL)\text{ forms a Hadamard triple}\}$$ constitutes a spectral eigenvalue set for $μ_{q,D}$, where $(q,\mathcal{D},L)$ is a Hadamard triple. And they prove for $s \in [0,\frac{\log \#\mathcal{D}}{\log q}]$, the corresponding spectra form a family of cardinality continuum. In this paper, we study Moran measures formed by random convolutions of finite Hadamard triples. Let $$μ=δ_{M_1^{-1}D_1}*δ_{M_2^{-1}D_2}*\cdots, \qquad M_k=q_1q_2\cdots q_k,$$ where the factors are produced from finitely many triples $\{(N_j,B_j,L_j):1\le j\le m\}$, $(ω_k)_{k=1}^{\infty}\in\{1,2,\ldots,m\}^{\mathbb N}$, and $n_k\in\mathbb N^+$, by setting $$ q_k=N_{ω_k}^{n_k},\qquad D_k=B_{ω_k},\qquad E_k=N_{ω_k}^{n_k-1}L_{ω_k}. $$ Assume a non-full-digit gap $$ρ:=\min_{1\le j\le m}\frac{N_j}{\#B_j}>1.$$ For the common Hadamard triple multiplier set $$\mathcal{T}_*:=\bigcap_{j=1}^m\{t\in\mathbb{Z}\setminus\{0\}:(N_j,B_j,tL_j)\text{ is a Hadamard triple}\},$$ Our main result is that, for every $$0\le s\le κ_ω:=\limsup_{R\to\infty}\frac{\sum_{r=1}^R\log \#D_r}{\sum_{r=1}^R\log q_r},$$ there exist continuum many countable sets $Λ\subset\mathbb{Z}$ such that $tΛ$ is a spectrum of $μ$ for every $t\in\mathcal{T}_*$ and $\dim_{Be}Λ=s$.

math.FA↗

arXiv2026

A Stable SBP-SAT FDTD Subgridding Method Without Region Split

A provably stable summation-by-parts simultaneous approximation term (SBP-SAT) finite-difference time-domain (FDTD) subgridding method without region split is proposed. By designing projection SBP operators tailored for embedded topological features and deriving the corresponding SAT boundary conditions, this approach guarantees long-time stability through discrete energy analysis. Unlike conventional SBP-SAT FDTD subgridding techniques that rely on aligned or multi-block configurations, the proposed method enables a direct coupling between an internal refined region and a single surrounding coarse-grid domain without introducing auxiliary blocks or causing domain fragmentation. Numerical results validate the efficiency, accuracy, and topological flexibility of the proposed method. Compared with existing multi-block SBP-SAT methods, this method effectively reduces computational complexity by minimizing SAT boundary conditions and improves calculation accuracy near grid interfaces.

cs.CE↗