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Weixiong Mai

Publications and source records attributed to Weixiong Mai.

14 recordsLinked to original sources
arXiv2025

Uncertainty principles for free metaplectic transformation and associated metaplectic operators

In this paper, we systematically investigate the Heisenberg-Pauli-Weyl uncertainty principle for free metaplectic transformation, as well as metaplectic operators. Specifically, we obtain two different types of the uncertainty principle for free metaplectic transformations in terms of the so-called phase derivative, one of which can be generalized to the $L^p$-case with $1\le p\le 2$. The obtained results are valid not only for free metaplectic transformations but also for general metaplectic operators. In particular, we point out that our results are closely related to those given in \cite{Dias-deGosson-Prata}, and the relationship should be new and not exactly given in the existing literature.

math.FA↗
arXiv2025

Paley-Wiener Theorems For Slice Regular Functions

We prove two theorems of Paley and Wiener in the slice regular setting. As an application, we can compute the reproducing kernel for the slice regular Paley-Wiener space, and obtain a related sampling theorem.

math.CV↗
arXiv2025

Paley-Wiener theorems for slice monogenic functions

In this paper, we prove some Paley-Wiener theorems for function spaces consisting of slice monogenic functions such as Paley-Wiener, Hardy and Bergman spaces. As applications, we can compute the reproducing kernel functions for the related function spaces.

math.CV↗
arXiv2023

Improved Caffarelli-Kohn-Nirenberg Inequalities and Uncertainty Principle

In this paper we prove some improved Caffarelli-Kohn-Nirenberg inequalities and uncertainty principle for complex- and vector-valued functions on $\mathbb R^n$, which is a further study of the results in \cite{Dang-Deng-Qian}. In particular, we introduce an analogue of "phase derivative" for vector-valued functions. Moreover, using the introduced "phase derivative", we extend the extra-strong uncertainty principle to cases for complex- and vector-valued functions defined on $\mathbb S^n,n\geq 2.$

math.AP↗
arXiv2023

On the Bergman kernel in weighted monogenic Bargmann-Fock spaces

In this paper, we study the Bergman kernel $B_φ(x,y)$ of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of $L^2$-estimate method and weighted subharmonic inequality for Clifford algebra are established. Consequently we show the existence of $B_φ(x,y)$ and then give some estimates on and off the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.

math.CV↗
arXiv2022

Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

math.DG↗
arXiv2022

On equidistribution theorem for multi-sequences of holomorphic line bundles

Given several sequences of Hermitian holomorphic line bundles $\{(L_{kp}, h_{kp})\}_{p=1}^{\infty}$, we establish the distribution of common zeros of random holomorphic sections of $L_{kp}$ with respect to singular measures. We also study the dimension growth for a sequence of pseudo-effective line bundles.

math.CV↗
arXiv2022

Metric Distributional Discrepancy in Metric Space

Independence analysis is an indispensable step before regression analysis to find out essential factors that influence the objects. With many applications in machine Learning, medical Learning and a variety of disciplines, statistical methods of measuring the relationship between random variables have been well studied in vector spaces. However, there are few methods developed to verify the relation between random elements in metric spaces. In this paper, we present a novel index called metric distributional discrepancy (MDD) to measure the dependence between a random element $X$ and a categorical variable $Y$, which is applicable to the medical image and genetic data. The metric distributional discrepancy statistics can be considered as the distance between the conditional distribution of $X$ given each class of $Y$ and the unconditional distribution of $X$. MDD enjoys some significant merits compared to other dependence-measures. For instance, MDD is zero if and only if $X$ and $Y$ are independent. MDD test is a distribution-free test since there is no assumption on the distribution of random elements. Furthermore, MDD test is robust to the data with heavy-tailed distribution and potential outliers. We demonstrate the validity of our theory and the property of the MDD test by several numerical experiments and real data analysis.

stat.ME↗
arXiv2021

On Monogenic Reproducing Kernel Hilbert Spaces of the Paley-Wiener Type

In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley-Wiener type, namely the Paley-Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels.

math.CV↗
arXiv2020

The Fourier Type Expansions on Tubes

In view of recent developments of the study of reproducing kernel Hilbert spaces, in particular with the context the Hardy spaces on tubes, aspects of rational approximation for functions of finite energy in several complex and several real variables are developed.

math.CV↗
arXiv2019

Fourier Spectrum Characterizations of Clifford $H^{p}$ Spaces on $\mathbf{R}^{n+1}_+$ for $1\leq p \leq \infty$

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq \infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is further the non-tangential boundary limit of some function in $H^p(\mathbf R^{n+1}_+),$ $1\leq p\leq \infty,$ if and only if $\hat{f}=χ_+\hat{f},$ where $χ_+(\underlineξ)=\frac{1}{2}(1+i\frac{\underline ξ}{|\underline ξ|}),$ where the Fourier transformation and the above relation are suitably interpreted (for some cases in the distribution sense). These results further develop the relevant context of Alan McIntosh. As a particular case of our results, the vector-valued Clifford Hardy space functions are identical with the conjugate harmonic systems in the work of Stein and Weiss. The latter proved the corresponding results in terms of the single integral form for the cases $1\leq p<\infty.$

math.CV↗
arXiv2016

Aveiro Method in Reproducing Kernel Hilbert Spaces Under Complete Dictionary

Aveiro Method is a sparse representation method in reproducing kernel Hilbert spaces (RKHS) that gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying RKHS. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro Method. To avoid those difficulties we propose an anew Aveiro Method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so called Pre-Orthogonal Greedy Algorithm (P-OGA) involving completion of a given dictionary. The new method is called Aveiro Method Under Complete Dictionary (AMUCD). The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition, bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro Method and the greedy algorithm.

math.CV↗