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A representation formula for slice regular functions over slice-cones in several variables

Xinyuan Dou, Guangbin Ren, Irene Sabadini

Abstract

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in $[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $τ_s$ to this cone. Slice regular functions can be defined on open sets in $\left(τ_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

A representation formula for slice regular functions over slice-cones in several variables

Abstract

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form . We define a cone in and we extend the slice-topology to this cone. Slice regular functions can be defined on open sets in and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative -algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

Paper Structure

This paper contains 16 sections, 35 theorems, 183 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transformation matrix
  4. The Moore-Penrose inverse
  5. Complex structure
  6. Weak slice-cone
  7. Weak slice regular functions
  8. Extension lemma
  9. Path-representation formula
  10. Weak slice regular functions over LSCS algebras
  11. LSCS algebras
  12. Weak slice regular functions over LSCS algebras
  13. The case of left alternative algebras
  14. Real alternative $*$-algebras, case $d=1$
  15. Clifford algebra $\mathbb{R}_{m}$ case for $d=1$
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $k,\ell,m\in\mathbb{N}_+$, $S,T\in\mathop{\mathrm{End}}\left(\mathbb{R}^{2n}\right)^{k\times\ell}$ and $U\in\mathop{\mathrm{End}}\left(\mathbb{R}^{2n}\right)^{\ell\times m}$. Then the following statements hold.

Theorems & Definitions (80)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 3.1
  • Definition 3.2
  • Example 3.4
  • ...and 70 more