On axially rational regular functions and Schur analysis in the Clifford-Appell setting

IF 1.4 3区 数学 Q1 MATHEMATICS
Daniel Alpay, Fabrizio Colombo, Antonino De Martino, Kamal Diki, Irene Sabadini
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引用次数: 0

Abstract

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in \({\mathbb {R}}^4\). The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized Cauchy–Kovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a role. The first one is the so-called slice operator, which extends holomorphic functions of one complex variable to slice hyperholomorphic functions of a quaternionic variable. The second operator is the Laplace operator in four real variables, that maps slice hyperholomorphic functions to axially regular functions. On the other hand, the generalized CK-extension gives a characterization of axially regular functions in terms of their restriction to the real line. In this paper we use these two extensions to define two notions of rational function in the regular setting. For our purposes, the notion coming from the generalized CK-extension is the most suitable. Our results allow to consider the Hardy space, Schur multipliers and their relation with realizations in the framework of Clifford-Appell polynomials. We also introduce two notions of regular Blaschke factors, through the Fueter theorem and the generalized CK-extension.

Abstract Image

论轴向有理正则函数和克利福德-阿佩尔环境中的舒尔分析
本文开始研究 Cauchy-Fueter 正四元数值函数的舒尔分析,即 Cauchy-Fueter 算子在 \({\mathbb {R}}^4\) 中的空解。本文方法的新颖之处在于我们考虑了轴正则函数,即所谓的克里福德-阿佩尔多项式所跨越的函数。这类函数自然产生于超复分析中两个著名的扩展结果:Fueter 映射定理和广义 Cauchy-Kovalevskaya (GCK) 扩展。这些结果允许人们从一个实变或复变的解析函数出发,获得轴正则函数。确切地说,在富特定理中,有两个算子在起作用。第一个是所谓的切片算子,它将一个复变函数的全纯函数扩展为一个四元变量的切片超全纯函数。第二个算子是四实变的拉普拉斯算子,它将切片超全貌函数映射为轴正则函数。另一方面,广义 CK 扩展给出了轴正则函数对实线的限制。在本文中,我们利用这两个扩展定义了正则环境中的两个有理函数概念。就我们的目的而言,来自广义 CK 扩展的概念是最合适的。我们的结果允许我们考虑哈代空间、舒尔乘数及其与克利福德-阿佩尔多项式框架中的实数的关系。我们还通过 Fueter 定理和广义 CK 扩展引入了正则布拉什克因子的两个概念。
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来源期刊
Analysis and Mathematical Physics
Analysis and Mathematical Physics MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.70
自引率
0.00%
发文量
122
期刊介绍: Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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