论文标题
S光谱上的阶2的多序分析函数演算
A polyanalytic functional calculus of order 2 on the S-spectrum
论文作者
论文摘要
Fueter定理提供了两个步骤，以构建轴向单基函数，即$ \ Mathbb {r}^4 $中的Cauchy-Riemann Operator的无溶液，由$ \ Mathcal {d} $表示。在第一步中，通过所谓的切片操作员将全态函数扩展到切片的超晶函数。在第二步中，通过将Laplace操作员分别以四个实际变量（$δ$）应用于切片的超酚形态函数来构建单基因函数。在本文中，我们使用拉普拉斯运算符的分解，即$δ= \ Mathcal {\ overline {d}} \ Mathcal {d} $来拆分上一个过程。从这种分裂中，我们获得了一类功能，该功能位于切片的超晶函数和轴向单基因函数集之间：订单2的轴向多芯片分析函数集，即$ \ MATHCAL {D}^2 $的null-Null分析。我们显示了此类功能的整体表示公式。获得的公式是定义$ s $ spectrum上关联的功能演算的基础。据作者所知，这是第一次考虑单基因分析功能积分。
The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $ \mathbb{R}^4$, denoted by $ \mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($Δ$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $Δ= \mathcal{\overline{D}} \mathcal{D}$ to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of $ \mathcal{D}^2$. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the $S$-spectrum. As far as the authors know, this is the first time that a monogenic polyanalytic functional calculus has been taken into consideration.
