切片超全纯函数的Corona问题

切片超全纯背景下的Fock空间

The Fock space in the slice hyperholomorphic setting

本文介绍并研究了切片超全纯背景下的Fock空间（又称Segal-Bargmann空间）的一些基本性质。我们讨论了四元数上的切片正则函数的情况，也讨论了值在Clifford代数上的切片单演函数的情况。在四元数的具体设置中，我们还引入了完整的Fock空间。本文可以看作是在四元数背景下研究无穷维分析的开端。

片超全纯广义压缩函数和正函数的实现

Realizations of slice hyperholomorphic generalized contractive and positive functions

在切片超全纯函数的背景下引入广义Schur函数和广义正函数，并研究了它们在相应再生核Pontryagin空间中的实现

切片超全纯背景下的Krein-Langer分解及相关问题

Krein-Langer factorization and related topics in the slice hyperholomorphic setting

我们研究了切片超全纯背景下Schur分析的各个方面。我们给出了两组结果：第一，我们给出了切片超全纯函数的谓词演算的新结果。特别地，我们介绍并研究了Riesz投影的一些性质。然后我们证明了一个Beurling-Lax型定理，即所谓的结构定理。一个关键的事实，允许证明我们的结果，是事实上，一个四元数线性算子的右谱和S-谱重合。最后，我们研究了切片超全纯广义Schur函数的Krein-Langer分解。Beurling-Lax型定理和Krein-Langer分解都是在四元数情形下用片超全纯以外的超全纯概念证明的具有深远意义的结果

切片超全纯Schur函数的边界插值

Boundary interpolation for slice hyperholomorphic Schur functions

在四元数背景下，提出并求解了一个边界Nevanlinna-Pick插值问题.给定非负实数$κ_1, \ldots, κ_N$，四元数$p_1, \ldots, p_N$所有的模$1$，使$2$ -领域确定的每个点不相交和$p_u \neq 1$的$u = 1,\ldots, N$，和四元数$s_1, \ldots, s_N$，我们希望找到一个片超全纯舒尔函数$s$，使$$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$和$$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\leκ_u,\quad {\rm for} \quad u=1,\ldots, N.$$我们的论点依赖于理论片超全纯函数和再生核希尔伯特空间。

关于因子化Fueter-Sce映射在切超复Cauchy核中的应用

On the application of the factorized Fueter-Sce map to the slice hyperholomorphic Cauchy kernel

本文研究了Fueter-Sce映射的最一般因子化，并将其应用于切超复函数的Cauchy核。

切片超全纯函数的Pontryagin de Branges Rovnyak空间

Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions

本文研究切片超全纯函数的再生核Hilbert空间和Pontryagin空间，它们是de Branges和Rovnyak引入的解析函数的希尔伯特空间的类似物.在本文的第一部分，我们专注于希尔伯特空间的情况下，并特别介绍了一个版本的哈代空间。然后定义了Blaschke因子和Blaschke乘积，并考虑了一个插值问题.在本文的第二部分，我们转向庞特里亚金空间的情况。我们首先证明了一些结果从理论的Pontryagin空间在四元数设置，特别是一个定理的Shmulyan稠密定义的压缩线性关系。然后研究了广义Schur函数和广义Carath'eodory函数的实现。

四元数切片正则函数与富埃特超全纯函数之间的联系

A connection between quaternionic slice regular and Fueter hyperholormorphic functions

研究四元数切片正则函数与富埃特超全纯函数联系，借助全局算子\(G\)、富埃特算子\(\mathcal D\)和拉普拉斯算子\(\Delta_{\mathbb{R}^{4}}\)的新关系得相关理论结果，还引入并研究了与单位球相关的超全纯函数子模。

球与半空间中的广义四元数Schur函数与Krein-Langer分解

Generalized quaternionic Schur functions in the ball and half-space and Krein-Langer factorization

本文在切片超全纯情形下证明了一个新的Krein-Langer分解定理，它比文[D. Alpay，F.科伦坡岛Sabadini，Krein-Langer分解和切片超全纯设置中的相关主题，J. Geom. Anal.，24（2014），843- 872]。我们对待的情况下，功能与$κ$负平方定义的子集的四元数单位球或子集的半空间的四元数与正实部。在证明我们的结果的一个重要工具是Schauder-Tychonoff定理和一个不变子空间定理的压缩庞特里亚金空间。

On the global operator and Fueter mapping theorem for slice polyanalytic functions

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral represe…

A quaternionic analogue of the Segal-Bargmann transform

The Bargmann-Fock space of slice hyperholomorphic functions is recently introduced by Alpay, Colombo, Sabadini and Salomon. In this paper, we reconsider this space and present a direct proof of its independence of the slice. We also introduce a quaternionic analogue of the classical Segal-Bargmann transform and discuss some of its basic properties. The explicit expression of its inverse is obtained and the connection to the left one-dimensional quaternionic Fourier transform is given.

S--polyregular Bargmann spaces

We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit formulas of their reproducing kernels are given and associated Segal--Bargmann transforms are also introduced and studied. The spectral description as special subspaces of $L^2$-eigenspaces of a second order differential operator involving the slice derivative is investigated.

超复分析中的 Fock 空间探究

Towards Fock Spaces in Hypercomplex Analysis

本章综述了从复分析到超复分析的 Fock 空间，重点讨论了四元数变量下的切片超全纯和多解析 Fock 空间，并通过 Fueter 映射定理建立其与经典理论的联系。

On the Bargmann-Fock-Fueter and Bergman-Fueter integral transforms

This paper deals with some special integral transforms of Bargmann-Fock type in the setting of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. The construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. The ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic setting are obtained in both the Fock…

通过矩阵 corona 问题研究切片全纯函数的 corona 问题

The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem

本文针对取值于任意 $n\geq2$ 的 Clifford 代数 $\mathbb R_n$ 的切片超全纯函数，通过结合矩阵版 corona 定理、Bezout 恒等式与 Carleson 型条件，提出新的切片乘积写法，通用证明了 corona 定理。

On a class of orientation-preserving maps of $\mathbb R^4$

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian $J_f$ of a slice regular function $f$ proving in particular that $\det(J_f)\geq0$, i.e. $f$ is orientation-preserving. We give a complete characterization …