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A unified approach to the Dirac fine structures on the $S$-spectrum and a connection with Jacobi polynomials

F. Colombo, A. De Martino, S. Pinton

TL;DR

This work develops a unified fine-structure framework for the $S$-spectrum in Clifford analysis by factorizing the Dirac-Laplacian $ abla_{n+1}^{h_n}$ into $D^eta abla_{n+1}^m$ and $ar{D}^eta abla_{n+1}^m$ (with $eta+m\, leq h_n$) and applying Fueter–Sce extension to connect slice hyperholomorphic and axially monogenic function spaces. It provides explicit integral representations for the resulting axially Analytic-Harmonic and Anti-Analytic-Harmonic functions, and derives series expansions of the kernels in terms of Jacobi polynomials, revealing a deep link between pseudo-Cauchy kernels and classical orthogonal polynomials. Building on these representations, the paper constructs $D$- and $ar{D}$-functional calculi on the $S$-spectrum for commuting paravector operators, generalizing Poly-Harmonic and Poly-Analytic calculi and establishing kernel-independence and analytic structure, with potential $H^ty$-extensions. Overall, the results anchor a robust operator-calculus framework on the $S$-spectrum with rich connections to Jacobi polynomials and poly-analytic structures, offering new avenues for spectral theory in Clifford analysis and related monogenic settings.

Abstract

This paper contributes to the recently introduced theory of fine structures on the $S$-spectrum. We study, in a unified way, the functional calculi for axially Poly-Analytic-Harmonic functions on the $S$-spectrum. Axially Poly-Analytic-Harmonic functions of type $(β, m)$, for $β, m \in \mathbb{N}_0$ belong to the kernel of the Dirac-Laplace operators $D^βΔ^m_{n+1}$ of type $(β, m)$ and contain as particular cases Poly-Analytic and Poly-Harmonic functions of axial type. By applying these operators to the Cauchy kernels $S^{-1}_L(s,x)$ of (left) slice hyperholomorphic functions, we obtain an integral representation for axially Poly-Analytic-Harmonic functions. We point out that the kernels $D^βΔ^m_{n+1}S^{-1}_L(s,x)$ have a remarkable connection with Jacobi polynomials. By replacing the paravector operator $T$ with commuting components in the kernels $D^βΔ^m_{n+1} S^{-1}_L(s,x)$, we obtain the associated resolvent operators. With these resolvent operators, denoted by $S^{-1}_{L, D^βΔ^m}(s,T)$, we define the associated functional calculi based on the $S$-spectrum and study their properties.

A unified approach to the Dirac fine structures on the $S$-spectrum and a connection with Jacobi polynomials

TL;DR

This work develops a unified fine-structure framework for the -spectrum in Clifford analysis by factorizing the Dirac-Laplacian into and (with ) and applying Fueter–Sce extension to connect slice hyperholomorphic and axially monogenic function spaces. It provides explicit integral representations for the resulting axially Analytic-Harmonic and Anti-Analytic-Harmonic functions, and derives series expansions of the kernels in terms of Jacobi polynomials, revealing a deep link between pseudo-Cauchy kernels and classical orthogonal polynomials. Building on these representations, the paper constructs - and -functional calculi on the -spectrum for commuting paravector operators, generalizing Poly-Harmonic and Poly-Analytic calculi and establishing kernel-independence and analytic structure, with potential -extensions. Overall, the results anchor a robust operator-calculus framework on the -spectrum with rich connections to Jacobi polynomials and poly-analytic structures, offering new avenues for spectral theory in Clifford analysis and related monogenic settings.

Abstract

This paper contributes to the recently introduced theory of fine structures on the -spectrum. We study, in a unified way, the functional calculi for axially Poly-Analytic-Harmonic functions on the -spectrum. Axially Poly-Analytic-Harmonic functions of type , for belong to the kernel of the Dirac-Laplace operators of type and contain as particular cases Poly-Analytic and Poly-Harmonic functions of axial type. By applying these operators to the Cauchy kernels of (left) slice hyperholomorphic functions, we obtain an integral representation for axially Poly-Analytic-Harmonic functions. We point out that the kernels have a remarkable connection with Jacobi polynomials. By replacing the paravector operator with commuting components in the kernels , we obtain the associated resolvent operators. With these resolvent operators, denoted by , we define the associated functional calculi based on the -spectrum and study their properties.
Paper Structure (9 sections, 41 theorems, 255 equations)

This paper contains 9 sections, 41 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminary results on Clifford algebras and function theories
  3. The Dirac fine structures
  4. Integral representation of the functions of the Dirac fine structures
  5. Series expansions of the kernels $\mathcal{Q}^{-\ell}_{c,s}(x)$ and Jacobi polynomials
  6. Functional calculi of the Dirac fine structures
  7. The $S$-functional calculus for $n$-tuples of operators
  8. The $D$ and $\overline{D}$ functional calculi on the $S$-spectrum
  9. Appendix

Key Result

Theorem 2.9

Let $U \subset \mathbb{R}^{n+1}$ be an open set and $I \in \mathbb{S}$. We assume that $f$ is a left slice hyperholomorphic function and $g$ is a right slice hyperholomorphic function in $U$. Furthermore, let $D_I \subset U \cap \mathbb{C}_I$ be an open and bounded subset of $\mathbb{C}_I$ with $\ov

Theorems & Definitions (95)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6: Slice hyperholomorphic functions (or slice monogenic functions)
  • Definition 2.7
  • Theorem 2.9
  • Proposition 2.10: Cauchy kernel series
  • Proposition 2.11
  • Definition 2.12: Slice hyperholomorphic Cauchy kernels
  • Definition 2.15: Slice Cauchy domain
  • ...and 85 more