arXiv subjects

Zunwei Fu

Publications and source records attributed to Zunwei Fu.

At least 19 recordsLinked to original sources

arXiv2026

Recent progress of Littlewood-paley Theory with chirp function

Littlewood--Paley theory is a fundamental tool for frequency localization, square-function control, and multiplier analysis, yet a systematic counterpart in the fractional Fourier transform (FrFT) setting has remained incomplete. We develop a unified FrFT Littlewood--Paley framework based on the observation that, for a fixed $\alpha\notin\pi\mathbb Z$, a broad class of FrFT-side operators are exact chirp conjugates of their classical Fourier counterparts through $$ M_{\alpha}f(x)=e^{i\pi |x|^2\cot\alpha}f(x). $$ Within this unified framework we present: the FrFT multiplier identity; Littlewood--Paley square-function estimates and the converse theorem; sharp dyadic interval decompositions; Marcinkiewicz and Mihlin--H"ormander multiplier results; maximal, rough square-function, and almost-orthogonality estimates; twisted dyadic martingale geometry; inhomogeneous Sobolev, Besov, and Triebel--Lizorkin descriptions; Calder\'on reproducing formulae; pullback spaces and FrFT Riesz--Bessel operators; BMO, Carleson, sharp-maximal, and Hardy-space; twisted product estimates, multilinear bounds, and a Kato--Ponce theorem; fractional order-shifting in Lipschitz spaces; and the classical limit and singular boundary laws for the fractional parameter. The recurring theme is that a large class of FrFT operators are exact chirp conjugates of their classical counterparts, so most estimates are inherited with the same constants after one time identification of the rescaled symbols.

math.FA↗

arXiv2026

Limit Properties at Critical Indices of Linear Canonical Riesz Potentials and Their Applications to Security of Multi-Image Encryption

In this article we introduce the linear canonical Riesz potential (for short, LCRP) and give its symbol in terms of linear canonical transforms. Driven by image processing, we establish the convergence/divergence of these LCRPs for different kinds of functions. Concretely, for grating functions, we prove that their classical Riesz potentials diverge, whereas their LCRP converge due to the key role of chirp functions. For the characteristic function ${\mathbf 1}_P$ of a convex polygon $P$, we show that the limit of its Riesz potential at any non-boundary point $\boldsymbol{x}$ equals ${\mathbf 1}_P(\boldsymbol{x})$, but its limit at the boundaries differ from ${\mathbf 1}_P$, while it is known that, for any Schwartz function $f$, the limit of its Riesz potential at any point $\boldsymbol{x}$ always equals $f(\boldsymbol{x})$. Based on these and the inverse operator of the LCRP (namely the linear canonical Laplacian operator), we propose an asymmetric cascaded LCRP method for the multi-image encryption and create an efficient and secure cryptosystem. Systematic security evaluations, including sensitivity, statistical, noise attack, and occlusion attack analyses, demonstrate its robustness and its security. Even for a single image, the proposed method is more efficient than the known encryption approach based on the fractional Riesz potential. The novelty of these results lies in that the convergence and the divergence of LCRTs at the critical indices, respectively, for ``good" Schwartz functions and for ``bad" discrete image functions essentially affect the security of image encryption and decryption.

cs.CR↗

arXiv2026

Twisted Multiparameter singular integrals -- real variable methods and applications, I

In this paper, we introduce a class of twisted multiparameter singular integrals on $\mathbb{R}^{2m}$, motivated by the Cauchy--Szeg\H{o} projections and the solving operators for $\bar{\partial}_b$ on a broad family of quadratic surfaces of higher codimension in $\mathbb{C}^n$. These surfaces are represented as suitable quotients of products of Heisenberg groups, a framework illustrated by Stein (Notices Amer. Math. Soc., 1998). While classical multiparameter product and flag theories are well-developed, Nagel, Ricci, and Stein observed a critical limitation: the class of product operators is not closed under passage to a quotient subgroup. To handle the geometric reduction that models these quotient structures, we take the first step in developing an adapted real-variable theory. We achieve this by introducing twisted tube systems and tube maximal functions, establishing a reproducing formula, Littlewood--Paley theory, a Journ\'e-type covering lemma, and atomic decompositions. As particular examples, we obtain twisted Fourier multipliers -- which emerge as novel, direction-sensitive, and anisotropic phase-shift converters with potential applications in signal and image processing.

math.CA↗

arXiv2026

Characterizations of Hardy spaces on tube domains over polyhedral cones

This paper is devoted to the equivalence of various characterizations of holomorphic $H^1$ Hardy spaces on tube domains over polyhedral cones. We establish a new iterated Poisson integral formula which reproduces holomorphic functions on such domains. However, this formula shows that holomorphic $H^1$ functions have boundary values in a new type of Hardy space of real variables on their Shilov boundaries $\mathbb{R}^n$, which cannot be treated by standard classical multi-parameter harmonic analysis. We overcome this difficulty by developing techniques suitably adapted in this setting. Using the iterated Poisson integral as our approximation to the identity, and employing a lifting technique, we introduce various notions of multi-parameter analysis adapted to tube domains, such as twisted rectangles, new non-tangential approach regions, non-tangential maximal functions and Littlewood-Paley type functions. All these notions exhibit new geometric features associated with polyhedral cones and involve hidden parameters, as in the flag setting. We develop the necessary multi-parameter tools to investigate these new Hardy spaces. In particular, we apply these tools to obtain equivalent characterizations of the holomorphic $H^1$ Hardy spaces on tube domains in terms of non-tangential maximal, Lusin-Littlewood-Paley area and Littlewood-Paley $g$-functions.

math.CV↗

arXiv2025

Refining Image Edge Detection via Linear Canonical Riesz Transforms

Combining the linear canonical transform and the Riesz transform, we introduce the linear canonical Riesz transform (for short, LCRT), which is further proved to be a linear canonical multiplier. Using this LCRT multiplier, we conduct numerical simulations on images. Notably, the LCRT multiplier significantly reduces the complexity of the algorithm. Based on these we introduce the new concept of the sharpness $R^{\rm E}_{\rm sc}$ of the edge strength and continuity of images associated with the LCRT and, using it, we propose a new LCRT image edge detection method (for short, LCRT-IED method) and provide its mathematical foundation. Our experiments indicate that this sharpness $R^{\rm E}_{\rm sc}$ characterizes the macroscopic trend of edge variations of the image under consideration, while this new LCRT-IED method not only controls the overall edge strength and continuity of the image, but also excels in feature extraction in some local regions. These highlight the fundamental differences between the LCRT and the Riesz transform, which are precisely due to the multiparameter of the former. This new LCRT-IED method might be of significant importance for image feature extraction, image matching, and image refinement.

math.FA↗

arXiv2023

Sharp constants for a class of linear integral operators on higher-dimensional product spaces

In this paper, we will study a class of linear integral operators with the nonnegative kernels on higher-dimensional product spaces, the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. As application, we obtain the sharp constant for linear Hilbert operator on Higher-dimensional product spaces.

math.FA↗

arXiv2023

Boundedness of the multilinear integral operators on Heisenberg group

In this paper, we will obtain the sharp constant for multilinear integral operator on Heisenberg group Lebesgue space which is based on the Stein-Weiss lemma, the boundedness for multilinear integral operator on Heisenberg group $A_p$ weighted Morrey space and the sharp constant for multilinear integral operator on Heisenberg group two power weight Morrey space.

math.CA↗

arXiv2023

Fractional Fourier Transforms Meet Riesz Potentials and Image Processing

Via chirp functions from fractional Fourier transforms, the authors introduce fractional Riesz potentials related to chirp functions, establish their relations with fractional Fourier transforms, fractional Laplace operators related to chirp functions, and fractional Riesz transforms related to chirp functions, and obtain their boundedness on rotation invariant spaces related to chirp functions. Finally, the authors give the numerical image simulation of fractional Riesz potentials related to chirp functions and their applications in image processing. The main novelty of this article is to propose a new image encryption method for the double phase coding based on the fractional Riesz potential related to chirp functions. The symbol of fractional Riesz potentials related to chirp functions essentially provides greater degrees of freedom and greatly makes the information more secure.

math.FA↗

arXiv2022

Convergence of fractional Fourier series on the torus and applications

In this paper, we introduce the fractional Fourier series on the fractional torus and study some basic facts of fractional Fourier series, such as fractional convolution and fractional approximation. Meanwhile, fractional Fourier inversion and Poisson summation formula are also given. We further discuss the relationship between the decay of fractional Fourier coefficients and the smoothness of a function. Using the properties of fractional Fejer kernel, the pointwise convergence of fractional Fourier series can be established. Finally, we present the applications of fractional Fourier series to fractional partial differential equations with periodic boundary condition. Moreover, we apply approximation methods on the fractional torus to recover the non-stationary signals.

math.FA↗

arXiv2021

Riesz transform associated with the fractional Fourier transform and applications in image edge detection

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of is connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform which is defined by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transforms, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

math.FA↗

arXiv2020

Signal Encryption Strategy based on Domain change of the Fractional Fourier Transform

This paper provides a double encryption algorithm that uses the lack of invertibility of the fractional Fourier transform (FRFT) on $L^{1}$. One encryption key is a function, which maps a ``good" $L^{2}$-signal to a ``bad" $L^{1}$-signal. The FRFT parameter which describes the rotation associated with this operator on the time-frequency plane provides the other encryption key. With the help of approximate identities, such as of the Abel and Gauss means of the FRFT established in \cite{CFGW}, we recover the encrypted signal on the FRFT domain. This design of an encryption algorithm seems new even when using the classical Fourier transform. Finally, the feasibility of the new strategy is verified by simulation and audio examples.

cs.IT↗

arXiv2020

Fractional Fourier transforms on $L^p$ and applications

This paper is devoted to the $L^p(\mathbb R)$ theory of the fractional Fourier transform (FRFT) for $1\le p < 2$. In view of the special structure of the FRFT, we study FRFT properties of $L^1$ functions, via the introduction of a suitable chirp operator. However, in the $L^1(\mathbb{R})$ setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion problem via approximation by suitable means, such as the fractional Gauss and Abel means. We also obtain the regularity of fractional convolution and results on pointwise convergence of FRFT means. Finally we discuss $L^p$ multiplier results and a Littlewood-Paley theorem associated with FRFT.

math.FA↗

arXiv2020

Exponentially Weighted l_2 Regularization Strategy in Constructing Reinforced Second-order Fuzzy Rule-based Model

In the conventional Takagi-Sugeno-Kang (TSK)-type fuzzy models, constant or linear functions are usually utilized as the consequent parts of the fuzzy rules, but they cannot effectively describe the behavior within local regions defined by the antecedent parts. In this article, a theoretical and practical design methodology is developed to address this problem. First, the information granulation (Fuzzy C-Means) method is applied to capture the structure in the data and split the input space into subspaces, as well as form the antecedent parts. Second, the quadratic polynomials (QPs) are employed as the consequent parts. Compared with constant and linear functions, QPs can describe the input-output behavior within the local regions (subspaces) by refining the relationship between input and output variables. However, although QP can improve the approximation ability of the model, it could lead to the deterioration of the prediction ability of the model (e.g., overfitting). To handle this issue, we introduce an exponential weight approach inspired by the weight function theory encountered in harmonic analysis. More specifically, we adopt the exponential functions as the targeted penalty terms, which are equipped with l2 regularization (l2) (i.e., exponential weighted l2, ewl_2) to match the proposed reinforced second-order fuzzy rule-based model (RSFRM) properly. The advantage of el 2 compared to ordinary l2 lies in separately identifying and penalizing different types of polynomial terms in the coefficient estimation, and its results not only alleviate the overfitting and prevent the deterioration of generalization ability but also effectively release the prediction potential of the model.

cs.LG↗

arXiv2020

Cauchy-Szeg\"o operator, quaternionic Siegel upper half space, commutator, weighted Morrey space

In the setting of quaternionic Heisenberg group $\mathscr H^{n-1}$, we characterize the boundedness and compactness of commutator $[b,\mathcal C]$ for the Cauchy--Szeg\"o operator $\mathcal C$ on the weighted Morrey space $L_w^{p,\,\kappa}(\mathscr H^{n-1})$ with $p\in(1, \infty)$, $\kappa\in(0, 1)$ and $w\in A_p(\mathscr H^{n-1}).$ More precisely, we prove that $[b,\mathcal C]$ is bounded on $L_w^{p,\,\kappa}(\mathscr H^{n-1})$ if and only if $b\in {\rm BMO}(\mathscr H^{n-1})$. And $[b,\mathcal C]$ is compact on $L_w^{p,\,\kappa}(\mathscr H^{n-1})$ if and only if $b\in {\rm VMO}(\mathscr H^{n-1})$.

math.CV↗

arXiv2019

Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions

We establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows the local version of commutators also has an independent interest.

math.FA↗

arXiv2018

Morrey meets Muckenhoupt: A note on Nakai's generalized Morrey spaces and applications

The goal of this paper is to extend Nakai's generalized Morrey spaces to a wider function class, the one-sided Muckenhoupt weighted case. Morrey matching Muckenhoupt enables us to study the weak and strong type boundedness of one-sided sublinear operators satisfying certain size conditions on the one-sided weighted Morrey spaces. We also establish one-sided Fefferman-Stein inequalities on one-sided weighted Morrey spaces in this paper. Meanwhile, the boundedness and compactness of Riemann-Liouville integral operators on locally one-sided weighted Morrey space are considered. As applications, we establish the existence and uniqueness of solutions to a Cauchy type problem associated with fractional differential equations.

math.FA↗

arXiv2016

Gradient Estimates via Rearrangements for Solutions of Some Schr\"odinger Equations

In this article, by applying the well known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schr\"odinger equations, under the weak regularity assumption on the boundary of domains. As applications, gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

math.AP↗

arXiv2014

Some one-sided estimates for oscillatory singular integrals

The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\in A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the oscillatory singular integral is given.

math.CA↗