Toward Fock Spaces in Hypercomplex Analysis

Kamal Diki ()
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Kamal Diki: Ghent University, Clifford Research Group, Department of Electronics and Information Systems, Faculty of Engineering and Architecture

Chapter 71 in Operator Theory, 2026, pp 2235-2276 from Springer

Abstract: 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.

Keywords: Fock space; Segal–Bargmann transform; Quaternions; Slice hyperholomorphic (polyanalytic) functions; Fueter regular functions; Poly-Fueter regular functions; 30G35; 30H20; 46E22; 46S05 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_102

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