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Towards Fock Spaces in Hypercomplex Analysis

Kamal Diki

TL;DR

This work surveys the extension of the Segal-Bargmann (Fock) framework from the classical complex setting to hypercomplex analysis, focusing on quaternionic slice hyperholomorphic and Fueter-regular function theories. It builds a comprehensive collection of quaternionic Fock spaces, including slice hyperholomorphic, slice polyanalytic, Cholewinski-Fock, Gaussian RBF, and $p$-Fock variants, and links them via the Fueter mapping theorem to Fueter-regular and poly-Fueter-regular spaces. Key contributions include explicit reproducing kernels, Segal-Bargmann transforms, and approximation results, plus a poly-Fock-Fueter construction that unifies true poly-Fock spaces with their slice counterparts. The framework lays a foundation for quaternionic quantum mechanics, signal processing in the hypercomplex setting, and further development of Clifford-analytic Bargmann-type transforms. Overall, the article advances a cohesive theory connecting hypercomplex Fock spaces across slice and Fueter perspectives with concrete kernels and transform mappings.

Abstract

The Bargmann-Fock space(or Fock space for short) is a fundamental example of reproducing kernel Hilbert spaces that has found fascinating applications across multiple fields of current interest, including quantum mechanics, time-frequency analysis, mathematical analysis, and stochastic processes. In recent years, there has been increased interest in studying counterparts of the Fock space and related topics in hypercomplex analysis. This chapter presents a survey exploring various aspects of the Fock space from complex to hypercomplex analysis. In particular, we discuss different Fock spaces recently introduced in the setting of slice hyperholomorphic and slice polyanalytic functions of a quaternionic variable. The connection between slice hyperholomorphic (polyanalytic) Fock spaces and the classical theory of Fueter regular and poly-Fueter regular functions is established via the Fueter mapping theorem and its polyanalytic extension. This chapter focuses on Fock spaces consisting of functions of a quaternionic variable, with a brief discussion of related works in the Clifford setting.

Towards Fock Spaces in Hypercomplex Analysis

TL;DR

This work surveys the extension of the Segal-Bargmann (Fock) framework from the classical complex setting to hypercomplex analysis, focusing on quaternionic slice hyperholomorphic and Fueter-regular function theories. It builds a comprehensive collection of quaternionic Fock spaces, including slice hyperholomorphic, slice polyanalytic, Cholewinski-Fock, Gaussian RBF, and -Fock variants, and links them via the Fueter mapping theorem to Fueter-regular and poly-Fueter-regular spaces. Key contributions include explicit reproducing kernels, Segal-Bargmann transforms, and approximation results, plus a poly-Fock-Fueter construction that unifies true poly-Fock spaces with their slice counterparts. The framework lays a foundation for quaternionic quantum mechanics, signal processing in the hypercomplex setting, and further development of Clifford-analytic Bargmann-type transforms. Overall, the article advances a cohesive theory connecting hypercomplex Fock spaces across slice and Fueter perspectives with concrete kernels and transform mappings.

Abstract

The Bargmann-Fock space(or Fock space for short) is a fundamental example of reproducing kernel Hilbert spaces that has found fascinating applications across multiple fields of current interest, including quantum mechanics, time-frequency analysis, mathematical analysis, and stochastic processes. In recent years, there has been increased interest in studying counterparts of the Fock space and related topics in hypercomplex analysis. This chapter presents a survey exploring various aspects of the Fock space from complex to hypercomplex analysis. In particular, we discuss different Fock spaces recently introduced in the setting of slice hyperholomorphic and slice polyanalytic functions of a quaternionic variable. The connection between slice hyperholomorphic (polyanalytic) Fock spaces and the classical theory of Fueter regular and poly-Fueter regular functions is established via the Fueter mapping theorem and its polyanalytic extension. This chapter focuses on Fock spaces consisting of functions of a quaternionic variable, with a brief discussion of related works in the Clifford setting.

Paper Structure

This paper contains 17 sections, 38 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Fock spaces in one complex variable
  3. Analytic Case
  4. Polyanalytic Case
  5. Function theories of a quaternionic variable
  6. Slice Hyperholomorphic and Slice Polyanalytic Functions
  7. Fueter Regular and Poly-Fueter Regular Functions
  8. Fock spaces of slice hyperholomorphic functions
  9. Quaternionic Slice Hyperholomorphic Fock space and Its Segal-Bargmann Transform
  10. Gaussian Radial Basis Function (RBF) Kernel of a Quaternionic Variable
  11. Quaternionic Slice Hyperholomorphic Cholewinski-Fock Space
  12. Quaternionic $p$-Fock spaces and approximation
  13. Fock spaces of slice polyanalytic functions
  14. The Fueter mapping theorem and quaternionic Fock spaces
  15. The Fock-Fueter Space
  16. ...and 2 more sections

Key Result

Theorem 2.3

The Fock space $\mathcal{F}(\mathbb{C})$ admits the following analytic characterization Moreover, for $f(z)= \sum_{k=0}^{\infty} a_kz^k$ and $g(z)= \sum_{k=0}^{\infty} b_kz^k$ in $\mathcal{F}(\mathbb{C})$, the inner product is given by

Theorems & Definitions (114)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 104 more