Dmitri Bryukhov
The main aim of this paper is to describe the most adequate generalization of the Cauchy-Riemann system fixing properties of classical functions in octonionic case. An octonionic generalization of the Laplace transform is introduced. Octonionic generalizations of the inversion transformation, the gamma function and the Riemann zeta-function are given.
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Hermitean Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitean monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitean monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskay…
As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a corner stone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis in euclidean space R^{2n} of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on R^{2n}, quaternionic Clifford analysis arose as a further refinement by introducing a so--called hypercomplex structure Q…
In this paper we study some new special functions that arise naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac-like systems in several complex variables. In particular we focus on Hermite polynomials, Bessel functions and generalized powers. We also derive a Vekua system for solutions of Hermitian systems in axially symmetric domains.
Riemann-Hilbert problems for bi-axially symmetric null-solutions to iterated perturbed Dirac equations in R^n
本文研究R^n双轴对称域上带Clifford代数值变系数的迭代扰动Dirac算子零解的Riemann-Hilbert边值问题，通过适配双轴对称的Almansi型分解与递归积分算子构造显式解，推广至扰动情形并推导Schwarz问题闭式解，建立高维高阶边值问题的统一框架。
The Fueter-Sce mapping and the Clifford-Appell polynomials
本文利用Fueter-Sce映射与广义Cauchy-Kovalevskaya延拓的联系，显式计算其作用并刻画值域与核，定义新乘积，研究相关空间，并推广得到高维多解析单演多项式。
We focus on the Clifford-algebra valued variable coefficients Riemann-Hilbert boundary value problems $\big{(}$for short RHBVPs$\big{)}$ for axially monogenic functions on Euclidean space $\mathbb{R}^{n+1},n\in \mathbb{N}$. With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz…
A local Cauchy integral formula for slice-regular functions
本文证明了无需轴向对称的四元数空间开子集边界上的切片正则函数柯西型积分公式，并给出局部形式，通过分解为两个轴向单演函数实现证明。
In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs.
We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. We define (p,q)-left- and right-monogenic functions by means of Dirac operators that factor a certain wave operator. We prove two different versions of Cauchy's integral formulas for these functions. The two formulas arise from dealing with singularities in distinct ways, and are inspired by the methods of [L, FL]. These results indicate the merit of these methods for dealing with singularities.
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。
In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal Clifford analysis. After introducing Clifford algebra valued surface- and volume-elements first a purely fermionic Cauchy formula is proven. Combining this formula with the already well-known bosonic Cauchy formula yields the general case. Here the integration over the boundary of a supermanifold is an integration over as well the even as the odd boundary (in a f…
In this paper we study two-sided (left and right) axially symmetric solutions of a generalized Cauchy-Riemann operator. We present three methods to obtain special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave integrals and Funk-Hecke's formula and via primitivation. Each of these methods is effective enough to generate all the polynomial solutions.
New Taylor and Laurent series of axially harmonic, Fueter regular and polyanalytic functions
本文在四元数框架下研究Dirac型S谱精细结构的函数理论，给出轴向调和、Fueter正则及轴向多解析函数的新级数展开式，建立对应函数空间的表示公式，并阐述其在算子理论中的应用。
On the Fueter-Sce theorem and Cauchy-Kovalevskaya extensions over alternative $\ast$-algebras
本文研究了在超复子空间上定义的广义部分切函数理论，探讨了Fueter-Sce定理和三种类型的Cauchy-Kovalevskaya扩展及其内在关系，建立了单调性、广义部分切单调性和单调性的理论联系。
On Boundary Problems for Regular Functions in Hypercomplex Analysis
本文综述了 Clifford 分析中正则函数边值问题的研究，涵盖广义 Cauchy 定理等理论基础、积分表示，以及 Clifford 代数下的边值问题与奇异积分方程。
Almansi-type theorems for slice-regular functions on Clifford algebras
本文提出 Clifford 代数上多项式及 slice-正则函数的 Almansi 型分解，并揭示其与 Dirac 算子的联系，推广了经典多调和函数结果。
We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra $\mathbb R_n$ of signature $(0,n)$ and slice-regular functions on $\mathbb R_n$. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the {spherical derivative} of a slice …
We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For each suitable choice of a real vector subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case, we get a volume Cauchy formula. In the Clifford algebra case, the choice of the paravector subspace R^(n+1) gives a volume Cauchy formula for slice monogenic functions.
In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.
The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collabo…
Dunkl approach to slice regular functions
本文通过Dunkl分析与Clifford代数的联系，提出了一种基于全纯函数构造单生成函数的新方法，揭示了切正则函数与Dunkl算子核之间的关系。
A notion of fractional slice monogenic functions with respect to a pair of real valued functions
本研究提出了基于欧几里得空间特定域上一对实值函数的克利福德代数值分数切片单演函数理论，其由对应黎曼-刘维尔型和卡普托型分数柯西-黎曼算子零解定义，介绍了该理论的基本要素与结果。
In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and $a$-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative Clifford analysis setting so far. We also prove an argument principle for octonionic monogenic functions for isolated zeroes and for non-isolated compact zero sets. In the isolated case we c…
The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) = \int_{S^n}G_\om(z)\bs n(\om)f(\om)\,dμ(\om)$, $z\in L_n.$ The inverse map $\tilde f\mapsto f$ is constructed here using the Cauchy-Hua formula for the Lie ball following the work of M. Morimoto \cite{Mori2}.
Generalized partial-slice monogenic functions: the octonionic case
本文将广义偏切单生成函数的研究从关联的克莱因代数扩展到非关联交替代数，如八元数，并探讨其基本性质与贡献。
This is (raw) lecture notes of the course read on 6th European intensive course on Complex Analysis (Coimbra, Portugal) in 2000. Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained. The framework is based on wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Reduced wavelet transform allows naturally describe in abstract term main objects of an analytical function theory: the Cauchy integral formula, the Hardy and Bergman spaces,…
This paper extends approach of recent author's paper devoted to special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media and new generalizations of the Cauchy-Riemann system in $\mathbb R^3$. Two families of generalizations of the Cauchy-Riemann system with variable coefficients in $\mathbb R^4$ are presented in the context of non-Euclidean geometry. Analytic models of a wide range of potential meridional vector fields in $\mathbb R^4$ are characterized using a family of Vekua type systems in cylindrical coordinates. The specifics of meridional fields al…
Fractional slice regular functions of a quaternionic variable
本文提出四元数变量分数阶切片正则函数，定义为分数阶Cauchy-Riemann算子的零解，并证明表示定理与分裂引理，讨论级数展开。
Based on the representation of a set of canonical operators on the lattice $h\mathbb{Z}^n$, which are Clifford-vector-valued, we will introduce new families of special functions of hypercomplex variable possessing $\mathfrak{su}(1,1)$ symmetries. The Fourier decomposition of the space of Clifford-vector-valued polynomials with respect to the ${\rm SO}(n)\times \mathfrak{su}(1,1)$-module gives rise to the construction of new families of polynomial sequences as eigenfunctions of a coupled system involving forward/backward discretizations $E_h^{\pm}$ of the Euler operator $E=\sum\limits_{j=1}^nx_…
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