Fock and Hardy spaces: Clifford-Appell case

Clifford代数值函数Hardy空间的Szegö度量及其几何性质

The Szegö metric associated to Hardy spaces of Clifford algebra valued functions and some geometric properties

类似于复变函数论，我们在超复变函数论的背景下引入了赛格度量，该超复变函数论处理在克利福德代数中取值的函数。特别是，我们正在处理的Clifford代数值的功能，消灭了欧几里德狄拉克运营商在$\mathbb{R}^{m+1}$。这些通常被称为单演函数。由于两个由共形映射相互关联的域上的单演函数的Hardy空间之间的等距性，广义Szegö度量在莫比乌斯变换下具有伪不变性。这个属性是至关重要的应用，以表明该度量的曲率总是负的有界域。此外，它允许我们建立，这个度量是完全的有界域。

Gilbert猜想与一种从Clifford分析出发构造八元数解析函数的新方法

Gilbert's conjecture and A new way to octonionic analytic functions from the clifford analysis

本文通过八元数代数显式构造旋量空间$\mathcal{R}_8$与Clifford代数$Cl_8$，对$\mathbb{R}^8$上半空间Clifford解析函数Hardy空间的Gilbert猜想给出肯定回答，为八元数解析函数理论提供结合性研究路径，并重新表述相关经典结果。

厄米克利福德分析中的Szeg $\ddot{o}$投影和矩阵希尔伯特变换

Szeg$\ddot{o}$ projection and matrix Hilbert transform in Hermitean Clifford analysis

研究了Hermite-Clifford分析中两个复Hermite-Dirac算子的同时零解问题，提出了矩阵希尔伯特变换，并证明了它满足经典分析和正交Clifford分析中的希尔伯特变换的类似性质.在此背景下，我们将为定义在偶数维欧几里得空间有界子域上的厄米单演函数的Hardy空间引入Szeg $\ddot{o}$投影算子，建立将Szeg $\ddot{o}$投影算子与定义在偶数维欧几里得空间有界子域上的厄米单演函数的Hardy空间上的Hardy投影算子紧密联系起来的Kerzman-Stein公式，利用Hardy投影算子及其伴随算子得到Szeg $\ddot{o}$投影算子。进一步给出了Hermite-Clifford分析中矩阵希尔伯特变换酉性的代数和几何刻画。

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 furth…

高阶Lipschitz类由多单演函数的Hardy分解

Hardy decomposition of higher order Lipschitz classes by polymonogenic functions

本文利用clifford型Cauchy算子将高阶Lipschitz函数分解为多单演函数的迹并求解相关Riemann-Hilbert问题，提出了复平面Hardy分解的多维锐化版本。

利用Clifford代数值调和函数对一阶Lipschitz函数的分解

Decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

本文借助双单演函数论中奇异积分算子对应的Hardy投影，解决了满足特定边值条件的分段Clifford代数值调和函数构造问题，推广了复平面上的经典Hardy分解结果。

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.

On a polyanalytic a approach to noncommutative de Branges-Rovnyak spaces and Schur analysis

In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers a…

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator $D=\sum_{j=1}^n\e_j\partial_{x_j}$ together with the Lie-algebraic representation of raising and lowering operators acting on the lattice $h\BZ^n$ is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership o…

解析函数空间与波空间--讲义

Spaces of Analytical Functions and Wavelets--Lecture Notes

这是2000年第六届欧洲复分析强化课程（葡萄牙科英布拉）的课程讲稿。我们的目的是描述一个一般的框架推广的复分析。因此，不同的概括的分类方案得到。该框架是基于小波（相干态）在巴拿赫空间产生的“可接受的”群表示。约化小波变换使得解析函数论的主要对象：柯西积分公式、Hardy空间和Bergman空间、柯西-黎曼方程和泰勒展开式能够用抽象的术语自然地描述。其中考虑的例子是经典的解析函数理论（一个复变数，几个复变数，Clifford分析，Segal-Bargmann空间）以及新的函数理论，这是在我们的框架内开发（函数理论的双曲型，Clifford版本的Segal-Bargmann空间）。我们也简要地讨论了算子理论（谓词演算）和量子力学的应用。