arXiv subjects

Wenchang Sun

Publications and source records attributed to Wenchang Sun.

At least 19 recordsLinked to original sources

arXiv2026

Pointwise Convergence of Schr\"{o}dinger Operators in Bessel Potential Spaces

We study the pointwise convergence of solutions to the free Schr\"{o}dinger equation with initial data in the Bessel potential spaces $L_s^p(\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the full range $1 \leq p < \infty$, and demonstrate via counterexamples that these indices are sharp for all $1 \leq p \leq 2$ in one dimension, as well as for $p=1$ or $p$ large enough in higher dimensions. The proofs rely on the high-dimensional stationary phase method.

math.AP↗

arXiv2026

Frame Sets and Zeros of Zak transforms of Extended Gaussians

Let $a,b,c\in\mathbb C$ with $\re(a)<0$, we show that the extended Gaussian $e^{ax^2+bx+c}$ has maximal frame set (i.e., its frame set consists of precisely all positive pairs $(\alpha,\beta)$ with $\alpha\beta<1$), and its Zak transform has a unique simple zero in the unit square $[0,1)^2$ (in particular, the zero is at the center of the unit square if $b=0$). These statements extend the same results of the usual Gaussian (the cases when $a<0$ and $b,c\in\mathbb R$), and add more instances to the observation that if a continuous Wiener function has maximal frame set, then its Zak transform has a unique simple zero in the unit square. The proof of the maximality of the frame set combines metaplectic representation with a classical density result of the standard Gaussian. The proof of the uniqueness of the zero relies on properties of the theta function.

math.CA↗

arXiv2026

STFT Phase Retrieval with Two Window Functions

In this paper, we consider the uniqueness of STFT phase retrieval with two window functions. We show that a complex-valued locally integrable nonseparable signal is uniquely determined up to a global phase by phaseless samples of its short time Fourier transforms with respect to two well-chosen window functions over countable parallel lines or certain lattices. Moreover, we give the optimal sampling interval for STFT phase retrieval with compactly supported window functions. For periodic locally integrable signals, we obtain a uniqueness result for STFT phase retrieval with sampled values over two parallel lines whose distance is an irrational multiple of the period. And for quasi-periodic signals, we obtain a similar result.

math.CA↗

arXiv2025

Sampling Density for Gabor Phase Retrieval

Gabor phase retrieval stands for recovering a square integrable function up to a global phase from absolute values of its Gabor transform. In this paper, we study Gabor phase retrieval from discrete samples. We consider three types of sampling sequences, which include square root lattices, square root sequences on two intersecting lines and on three parallel lines respectively. In all cases we give the optimal sampling density for a sequence to do Gabor phase retrieval.

math.CA↗

arXiv2025

On the rate of convergence for Landau type Schr\"odinger Operators

We study the pointwise convergence of Landau type Schr\"odinger operators within the fractional Sobolev space $W^{s,p}(\mathbb R)$. Our results extend those established by Bailey (Rev. Mat. Iberoam., 29 (2): 531-546, 2013) and Yuan, Zhao and Zheng (Nonlinear Anal., 208: Paper No. 112312, 28, 2021). Furthermore, we also analyze the convergence rate of Landau type Schr\"odinger operators along curves and derive a sharp result for the case of convergence along vertical lines.

math.AP↗

arXiv2024

Nonuniform Sobolev Spaces

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the classical ones. These spaces arise naturally in the study of certain PDEs. For instance, we illustrate that nonuniform fractional Sobolev spaces are useful in the study of local estimates for solutions of heat equations and the convergence of Schr\"odinger operators. In this work we extend recent advances on local energy estimates for solutions of heat equations and the convergence of Schr\"odinger operators to nonuniform fractional Sobolev spaces.

math.AP↗

arXiv2020

Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective

We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{\textbf{i}α\cdot}φ), α\neq0,$ is impossible even though the real-valued function $φ$ enjoys the full spark property (so does $e^{\textbf{i}α\cdot}φ$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $ϕ$ such that the PLS of nonseparable causal (NC) signals in $V(ϕ)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability $1$, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability $1$, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.

cs.IT↗

arXiv2020

Hardy-Littlewood-Sobolev Inequality on Mixed-Norm Lebesgue Spaces

We consider the Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. We give a complete characterization of indices $\vec p$ and $\vec q$ such that the Riesz potential is bounded from $L^{\vec p}$ to $L^{\vec q}$, including all the endpoint cases. As a result, we get the mixed-norm Hardy-Littlewood-Sobolev inequality.

math.CA↗

arXiv2020

Extension of Multilinear Fractional Integral Operators to Linear Operators on Lebesgue Spaces with Mixed Norms

In [C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6(1):1-15, 1999], the following type of multilinear fractional integral \[ \int_{\mathbb{R}^{mn}} \frac{f_1(l_1(x_1,\ldots,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots,x_m,x))}{(|x_1|+\ldots+|x_m|)^λ} dx_1\ldots dx_m \] was studied, where $l_i$ are linear maps from $\mathbb{R}^{(m+1)n}$ to $\mathbb{R}^n$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $L^{p_1}\times \ldots \times L^{p_{m+1}}$ to $L^q$ when the indices satisfy the homogeneity condition. In this paper, we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed-norm Lebesgue space $L^{\vec p}$ which contains $L^{p_1}\times \ldots \times L^{p_{m+1}}$ as a subset. Under less restrictions on the linear maps $l_i$, we give a complete characterization of the indices $\vec p$, $q$ and $λ$ for which such an operator is bounded from $L^{\vec p}$ to $L^q$. And for $m=1$ or $n=1$, we give necessary and sufficient conditions on $(l_1, \ldots, l_{m+1})$, $\vec p=(p_1,\ldots, p_{m+1})$, $q$ and $λ$ such that the operator is bounded.

math.CA↗

arXiv2020

Bilinear Fractional Integral Operators

We study the bilinear fractional integral considered by Kenig and Stein, where linear combinations of variables with matrix coefficients are involved. Under more general settings, we give a complete characterization of the corresponding parameters for which the bilinear fractional integral is bounded from $L^{p_1}(\mathbb R^{n_1}) \times L^{p_2}(\mathbb R^{n_2})$ to $L^q(\mathbb R^m)$.

math.CA↗

arXiv2019

Bloom Type Inequality: The Off-diagonal Case

In this paper, we establish a representation formula for fractional integrals. As a consequence, for two fractional integral operators $I_{λ_1}$ and $I_{λ_2}$, we prove a Bloom type inequality \begin{align*} \mbox{\hbox to 8em{}}& \hskip -8em \left\|\big[I_{λ_1}^1,\big[b,I_{λ_2}^2\big]\big] \right\|_{L^{p_2}(L^{p_1})(μ_2^{p_2}\timesμ_1^{p_1})\rightarrow L^{q_2}(L^{q_1})(σ_2^{q_2}\timesσ_1^{q_1})} % \\ %& \lesssim_{\substack{[μ_1]_{A_{p_1,q_1}(\mathbb R^n)},[μ_2]_{A_{p_2,q_2}(\mathbb R^m)} \\ [σ_1]_{A_{p_1,q_1}(\mathbb R^n)},[σ_2]_{A_{p_2,q_2}(\mathbb R^m)}}} \|b\|_{\BMO_{\pro}(ν)}, \end{align*} where the indices satisfy $1<p_1<q_1<\infty$, $1<p_2<q_2<\infty$, $1/q_1+1/p_1'=λ_1/n$ and $1/q_2+1/p_2'=λ_2/m$, the weights $μ_1,σ_1 \in A_{p_1,q_1}(\mathbb R^n)$, $μ_2,σ_2 \in A_{p_2,q_2}(\mathbb R^m)$ and $ν:=μ_1σ_1^{-1}\otimes μ_2σ_2^{-1}$, $I_{λ_1}^1$ stands for $I_{λ_1}$ acting on the first variable and $I_{λ_2}^2$ stands for $I_{λ_2}$ acting on the second variable, $\BMO_{\rm{prod}}(ν)$ is a weighted product $\BMO$ space and $L^{p_2}(L^{p_1})(μ_2^{p_2}\timesμ_1^{p_1})$ and $ L^{q_2}(L^{q_1})(σ_2^{q_2}\timesσ_1^{q_1}) $ are mixed-norm spaces.

math.CA↗

arXiv2019

Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed Lebesgue space

The Mixed Lebesgue space is a suitable tool for modelling and measuring signals living in time-space domains. And sampling in such spaces plays an important role for processing high-dimensional time-varying signals. In this paper, we first define reproducing kernel subspaces of mixed Lebesgue spaces. Then, we study the frame properties and show that the reproducing kernel subspace has finite rate of innovation. Finally, we propose a semi-adaptive sampling scheme for time-space signals in a reproducing kernel subspace, where the sampling in time domain is conducted by a time encoding machine. Two kinds of timing sampling methods are considered and the corresponding iterative approximation algorithms with exponential convergence are given.

cs.IT↗

arXiv2018

Iterated and Mixed Weak Norms with Applications to Geometric Inequalities

In this paper, we consider a new weak norm, iterated weak norm in Lebesgue spaces with mixed norms. We study properties of the mixed weak norm and the iterated weak norm and present the relationship between the two weak norms. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for both weak norms. We study the convergence in truncated norm, which is a substitution of the convergence in measure for mixed Lebesgue spaces. And we give a characterization of the convergence in truncated norm. We show that Hölder's inequality is not always true on mixed weak spaces and we give a complete characterization of indices which admit Hölder's inequality. As applications, we establish some geometric inequalities related to fractional integration in mixed weak spaces and in iterated weak spaces which essentially generalize the Hardy-Littlewood-Sobolev inequality.

math.FA↗

arXiv2018

Nonuniform Sampling for Random Signals Bandlimited in the Linear Canonical Transform Domain

In this paper, we mainly investigate the nonuniform sampling for random signals which are bandlimited in the linear canonical transform (LCT) domain. We show that the nonuniform sampling for a random signal bandlimited in the LCT domain is equal to the uniform sampling in the sense of second order statistic characters after a pre-filter in the LCT domain. Moreover, we propose an approximate recovery approach for nonuniform sampling of random signals bandlimited in the LCT domain. Furthermore, we study the mean square error of the nonuniform sampling. Finally, we do some simulations to verify the correctness of our theoretical results.

eess.SP↗

arXiv2018

Uncertainty Principles Associated with the Offset Linear Canonical Transform

As a time-shifted and frequency-modulated version of the linear canonical transform (LCT), the offset linear canonical transform (OLCT) provides a more general framework of most existing linear integral transforms in signal processing and optics. To study simultaneous localization of a signal and its OLCT, the classical Heisenberg's uncertainty principle has been recently generalized for the OLCT. In this paper, we complement it by presenting another two uncertainty principles, i.e., Donoho-Stark's uncertainty principle and Amrein-Berthier-Benedicks's uncertainty principle, for the OLCT. Moreover, we generalize the short-time LCT to the short-time OLCT. We likewise present Lieb's uncertainty principle for the short-time OLCT and give a lower bound for its essential support.

eess.SP↗

arXiv2017

Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces

We study phaseless sampling in spline spaces generated by B-splines with arbitrary knots. For real spline spaces, we give a necessary and sufficient condition for a sequence of sampling points to admit a local phase retrieval of any nonseparable function. We also study phaseless sampling in complex spline spaces and illustrate that phase retrieval is impossible in this case. Nevertheless, we show that phaseless sampling is possible. For any function $f$ in a complex spline space, no mater it is separable or not, we show that $|f(x)|^2$ is uniquely determined and can be recovered linearly from its sampled values at a well chosen sequence of sampling points. We give necessary and sufficient conditions for such sequences.

math.FA↗

arXiv2017

Frame Phase-retrievability and Exact phase-retrievable frames

An exact phase-retrievable frame $\{f_{i}\}_{i}^{N}$ for an $n$-dimensional Hilbert space is a phase-retrievable frame that fails to be phase-retrievable if any one element is removed from the frame. Such a frame could have different lengths. We shall prove that for the real Hilbert space case, exact phase-retrievable frame of length $N$ exists for every $2n-1\leq N\leq n(n+1)/2$. For arbitrary frames we introduce the concept of redundancy with respect to its phase-retrievability and the concept of frames with exact PR-redundancy. We investigate the phase-retrievability by studying its maximal phase-retrievable subspaces with respect to a given frame which is not necessarily phase-retrievable. These maximal PR-subspaces could have different dimensions. We are able to identify the one with the largest dimension, which can be considered as a generalization of the characterization for phase-retrievable frames. In the basis case, we prove that if $M$ is a $k$-dimensional PR-subspace, then $|supp(x)| \geq k$ for every nonzero vector $x\in M$. Moreover, if $1\leq k< [(n+1)/2]$, then a $k$-dimensional PR-subspace is maximal if and only if there exists a vector $x\in M$ such that $|supp(x) | = k$.

math.FA↗

arXiv2017

Local and Global Phaseless Sampling in Real Spline Spaces

We study the recovery of functions in real spline spaces from unsigned sampled values. We consider two types of recovery. The one is to recover functions locally from finitely many unsigned samples. And the other is to recover functions on the whole line from infinitely many unsigned samples. In both cases, we give characterizations for a sequence of distinct points to be a phaseless sampling sequence, at which any nonseparable function is determined up to a sign on an interval or on the whole line by its unsigned sampled values. Moreover, for the case of local recovery, we also study the almost phaseless sampling and give a necessary and sufficient condition for a sequence of points to admit local recovery for almost all functions.

math.FA↗