Almansi-type Decomposition for Slice Regular Functions in Several Quaternionic Variables

Core Concepts

This paper introduces a novel Almansi-type decomposition for slice regular functions in several quaternionic variables, providing explicit formulas and exploring its applications in proving key theorems and deriving important formulas in quaternionic analysis.

Abstract

• Bibliographic Information: Binosi, G. (2024). Almansi-type decomposition for slice regular functions of several quaternionic variables. arXiv preprint arXiv:2209.06072v4.

• Research Objective: This paper aims to extend the Almansi-type decomposition for slice regular functions from one quaternionic variable to several variables and explore its applications in quaternionic analysis.

• Methodology: The paper utilizes the stem function approach to slice regularity and develops an explicit formula for the decomposition. It then applies this decomposition to prove existing theorems, such as Fueter's Theorem, and derive new results, including mean value and Poisson formulas for slice regular functions.

• Key Findings:

  • The paper establishes 2ⁿ distinct and unique Almansi-type decompositions for any slice function with a domain in Hⁿ, where n is the number of quaternionic variables.
  • Each component of the decomposition is explicitly given, uniquely determined, and exhibits desirable properties like harmonicity and circularity in specific variables.
  • The decomposition provides alternative proofs for Fueter's Theorem in Hⁿ and the biharmonicity of slice regular functions.
  • The paper derives mean value and Poisson formulas for slice regular functions using the decomposition.

• Main Conclusions: The Almansi-type decomposition for slice regular functions in several quaternionic variables is a powerful tool for studying these functions and their properties. It offers a new perspective on existing results and facilitates the derivation of new formulas in quaternionic analysis.

• Significance: This research significantly contributes to the field of quaternionic analysis by providing a new framework for understanding and manipulating slice regular functions in several variables. The explicit formulas and derived results have potential applications in various areas where quaternionic analysis is employed.

• Limitations and Future Research: The paper focuses on slice regular functions defined on circular sets. Future research could explore extending the Almansi-type decomposition to more general domains or other classes of hypercomplex functions. Additionally, investigating the applications of this decomposition in specific areas like mathematical physics or engineering could be promising.

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Almansi-type decomposition for slice regular functions of several quaternionic variables

by Giul... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2209.06072.pdf

Deeper Inquiries

How might this Almansi-type decomposition be applied to solve specific problems in applied mathematics or physics that involve quaternionic variables?

The Almansi-type decomposition for slice regular functions of several quaternionic variables, as presented in the paper, holds significant promise for various applications in applied mathematics and physics. Here are some potential avenues:
1. Solving Partial Differential Equations:

Simplification through Harmonicity: The decomposition allows expressing a slice regular function in terms of components that are harmonic with respect to certain variables. This property can be particularly useful when solving PDEs involving the Laplacian operator. By decomposing the solution function, one can potentially reduce the complexity of the problem and obtain solutions in terms of simpler harmonic functions.
Boundary Value Problems: In fields like electromagnetism and fluid dynamics, quaternions are often employed to represent physical quantities. The Almansi-type decomposition, with its explicit formulas for components, could be valuable in solving boundary value problems. The circularity and harmonicity properties of the components might simplify the task of satisfying boundary conditions.
2. Signal and Image Processing:

Quaternion-valued Signals: Quaternions are increasingly used in signal processing, particularly for color image processing where they can represent color channels effectively. The Almansi-type decomposition could lead to novel filtering techniques or feature extraction methods for quaternion-valued signals. Decomposing a signal into components with specific properties might help isolate different aspects of the signal for analysis.
Image Transformations: The decomposition could inspire new quaternion-based image transformations. By manipulating the individual components of an image's quaternion representation, one might achieve specific image processing effects, such as edge detection or noise reduction, in a more efficient or targeted manner.
3. Quantum Mechanics:

Quaternion Quantum Mechanics: There have been attempts to formulate quantum mechanics using quaternions. While not mainstream, the Almansi-type decomposition might find applications in this area. The decomposition's ability to separate a function into components with distinct properties could be relevant for analyzing quantum states or operators within a quaternionic framework.
4. Numerical Methods:

Developing Specialized Numerical Schemes: The explicit formulas for the components in the Almansi-type decomposition could be leveraged to develop specialized numerical methods for problems involving slice regular functions. These methods could exploit the properties of the components, such as harmonicity, to achieve better accuracy or efficiency compared to more general-purpose numerical schemes.
These are just a few potential applications. As research in quaternionic analysis progresses, the Almansi-type decomposition is likely to find even more uses in diverse areas of applied mathematics and physics.

Could there be alternative decompositions for slice regular functions with potentially different properties and applications?

It's certainly plausible that alternative decompositions for slice regular functions exist, each offering unique properties and potential applications. The paper focuses on a specific Almansi-type decomposition based on spherical values and derivatives, but other approaches could be explored:
1. Different Basis Functions:

Polynomial Bases: Instead of the monomial basis used in the paper's decomposition, one could investigate decompositions using other polynomial bases, such as orthogonal polynomials on the quaternionic space. This might lead to decompositions where the components have desirable properties related to orthogonality or approximation theory.
Special Function Bases: Exploring decompositions based on special functions, like quaternion-valued generalizations of Bessel functions or spherical harmonics, could be fruitful. These special functions often arise in solutions to specific PDEs, and using them as a basis for decomposition might provide insights into the behavior of slice regular functions in those contexts.
2. Operator-Based Decompositions:

Alternative Operators: Instead of relying solely on spherical values and derivatives, one could investigate decompositions based on other operators that have meaningful actions on slice regular functions. This could involve differential operators, integral operators, or combinations thereof.
Spectral Decompositions: If suitable operators with a well-defined spectrum can be identified, spectral decompositions of slice regular functions might be possible. This would involve expressing a function as a sum or integral of eigenfunctions of the chosen operator.
3. Geometric Decompositions:

Slice-Preserving and Slice-Rotating Components: The paper mentions slice-preserving functions. It might be interesting to explore decompositions where one component is slice-preserving while the other has a specific action on the slices, such as rotating them. This could be relevant for applications where the geometry of the quaternionic space plays a crucial role.
4. Decompositions Tailored to Applications:

Application-Specific Properties: For particular applications, it might be beneficial to seek decompositions where the components possess properties directly relevant to the problem at hand. For example, in signal processing, one might desire components that correspond to different frequency bands or time-frequency characteristics.
Exploring these alternative decompositions could enrich the theory of slice regular functions and uncover new connections with other areas of mathematics. The properties of each decomposition would dictate its suitability for specific applications.

What are the implications of this decomposition for the development of numerical methods in quaternionic analysis?

The Almansi-type decomposition presented in the paper has several important implications for the development of numerical methods in quaternionic analysis:
1. Exploiting Harmonicity for Efficient Solvers:

Harmonic Function Solvers: The decomposition allows expressing slice regular functions in terms of components that are harmonic with respect to certain quaternionic variables. This opens the door to leveraging existing, well-established numerical methods for solving Laplace's equation in the context of quaternionic analysis.
Reduced Computational Cost: By working with harmonic components, numerical solvers can potentially benefit from reduced computational complexity. Solving Laplace's equation is generally less computationally demanding than solving more general PDEs that slice regular functions might satisfy.
2. Tailored Discretization Schemes:

Slice-Based Discretization: The decomposition's structure, involving spherical values and derivatives, suggests the potential for developing numerical schemes that discretize the quaternionic space in a slice-by-slice manner. This could be advantageous for problems with specific symmetries or where the behavior of the function on different slices is of particular interest.
Adapting Existing Methods: Numerical methods for complex analysis, such as those based on Cauchy integral formulas or conformal mapping, could potentially be adapted to the quaternionic setting by leveraging the Almansi-type decomposition. The decomposition might provide a bridge between complex and quaternionic numerical techniques.
3. Handling Boundary Conditions:

Simplified Boundary Treatment: The explicit formulas for the components in the decomposition might simplify the treatment of boundary conditions in numerical methods. The circularity and harmonicity properties of the components could be exploited to construct numerical solutions that satisfy boundary conditions more accurately or efficiently.
4. Developing New Basis Functions:

Quaternion-Specific Basis: The decomposition itself suggests a potential set of basis functions for representing slice regular functions numerically. By discretizing the components of the decomposition, one could obtain a set of basis functions specifically designed for approximating slice regular functions.
5. Error Analysis and Convergence:

Analyzing Convergence: The decomposition could be a valuable tool for analyzing the convergence properties of numerical methods for slice regular functions. By understanding how the decomposition behaves under discretization, one can gain insights into the accuracy and stability of numerical schemes.
Overall, the Almansi-type decomposition provides a new framework for approaching numerical problems in quaternionic analysis. By exploiting the properties of the decomposition and its components, researchers can develop more efficient, accurate, and specialized numerical methods for solving PDEs, approximating functions, and tackling other computational challenges in this field.