The Bergman-Sce transform for slice monogenic functions

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n-tuples of commuting operators (called ℱ-functional calculus) based on a new notion of spectrum, called ℱ-spectrum, for the n-tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ℱ-functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.

In a recent paper [Trans. Amer. Math. Soc. 378 (2025), pp. 851–883], the concept of generalized partial-slice monogenic (or regular) function was introduced over Clifford algebras. The present paper shall extend the study of generalized partial-slice monogenic functions from the associative case of Clifford algebras to non-associative alternative algebras, such as octonions. The new class of functions encompasses the regular functions [Rend. Sem. Mat. Univ. Padova 50 (1973), pp. 251–267] and slice regular functions [Rocky Mountain J. Math. 40 (2010), no. 1, 225–241] over octonions, indeed both appear in the theory as special cases. In the non-associative setting of octonions, we shall develop some fundamental properties such as identity theorem, Representation Formula, Cauchy (and Cauchy-Pompeiu) integral formula, maximum modulus principle, Fueter polynomials, Taylor series expansion. As a complement, the paper also introduces and discusses the notion of generalized partial-slice (and regular) functions. Although the study is limited to the case of octonions, it is clear from the statements and the arguments in the proofs that the results hold more in general in real alternative algebras equipped with a notion of conjugation.

In Bures et al. (Elements of quaternionic analysis and Radon transform, 2009), the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on \({\mathbb {R}}^{n+1}\) using the Radon and dual Radon transforms. The main aim of this paper is to further develop this approach. In fact, the Radon transform for functions with values in the Clifford algebra \({\mathbb {R}}_n\) is mapping solutions of the generalized Cauchy–Riemann equation, i.e., monogenic functions, to a parametric family of holomorphic functions with values in \({\mathbb {R}}_n\) and, analogously, the dual Radon transform is mapping parametric families of holomorphic functions as above to monogenic functions. The parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so-called slice monogenic functions. An important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms.

After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular functions, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function f:Ω→H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\Omega \\rightarrow {\\mathbb {H}}$$\\end{document} a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent f better and better and allow to prove increasingly stronger results.

Special Bi-Axial Monogenic Functions

Contragenic functions are defined to be reduced-quaternion-valued harmonic functions which are orthogonal to all monogenic and antimonogenic functions in the $$L^2$$ L 2 norm of a given domain. The parallelism between the spaces of contragenic functions in the interior and exterior of the unit sphere in $$\mathbb {R}^3$$ R 3 is described in detail. Bergman reproducing kernels for the spaces of contragenic functions are given, mirroring the corresponding kernels for the spaces of vector parts of monogenic functions. Numerical examples are given showing the accuracy of truncations of the integral kernels. A striking duality is observed between the basic interior contragenic functions and the vector parts of exterior monogenic functions, and vice versa.

Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), Gonzalez Cervantes (Complex Anal Oper Theory 13:2527–2539, 2019) and Gonzalez Cervantes and Gonzalez-Campos (Complex Var Elliptic Equ 65:1–10, 2020, https://doi.org/10.1080/17476933.2020.1738410) [4, 13, 14], respectively. The aim of this work is to show: some relationships between G and the composition operator with the conformal mappings, a conformal covariance property of G along with its interpretations in terms of a covariant functor, all consequences of these facts for the slice regular functions, a Leibnitz rule associated to the operator G and a characterization of the real Components of slice regular functions in terms of a Non-constant Coefficient second order differential equation.

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{gensalsto} and developed in \cite{AAtwistor}. To any slice-polynomial function $P$ we associate its {\em companion} $P^\vee$ and its {\em extension} to the real axis $P_\mathbb{R}$, that are quaternionic functions naturally related to $P$. Then, using the theory of twistor spaces, we are able to show that for any quaternion $q$ the {cardinality of simultaneous} pre-images of $q$ via $P$, $P^\vee$ and $P_\mathbb{R}$ is generically constant, giving a notion of degree. With the brand new tool of slice-polynomial functions, we {compute} the twistor discriminant locus of a cubic scroll $\mathcal{C}$ in $\mathbb{CP}^3$ and we conclude by giving some qualitative results on the complex structures induced by $\mathcal{C}$ via the twistor projection.

This works introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green's functions.

Geometric function theory over quaternionic slice domains

Its significance for complex analysis lies in the fact that if N ƒ G dom D solves O(Nf) = ƒ and df = 0, then u — d Nf is the unique solution of du = ƒ which is orthogonal to kerd. J. J. Kohn has established existence and regularity properties of the solution operator iV, giving the solution of minimal norm—the so called <9-Neumann operator—in case D is strictly pseudoconvex [5], and in more general cases as well [6]. The proofs are based on a priori estimates in L-Sobolev spaces, and therefore they do not give any explicit information about the kernels of N or d N. In recent years there has been much interest in finding more explicit and concrete representations of the abstractly defined operators N and d N (see [2, 3, 9, 10, 12]). In [7] we began to study d N by using the calculus of Cauchy-Fantappie kernels in C n , in analogy to the work of Kerzman and Stein [4] and Ligocka [8] for the Szego, respectively, Bergman kernel; in contrast to the scalar case, the incompatibility of the Euclidean metric with the complex geometry of the boundary of D turned out to be a major obstruction in the general case. In the present paper we overcome this obstruction by generalizing the results in [7] to arbitrary Hermitian manifolds; this enables us to then introduce a special Levi metric—similar to the one in [2]—and to establish the required symmetry properties of the kernels. Our main result gives a new and completely explicit integral representation of the principal part of d N on

In this paper we define a product for monogenic functions derived from Fischer’s decomposition. Although this product can be defined for all monogenic polynomials, it can be canonically defined only for a special subclass of polynomials. The definition of the product can be generalized to the setting of monogenic tensors.

In this paper, starting from a function analytic in a neighborhood of the unit disk and based on Bessel functions, we construct a family of generalized multivariate sinc functions, which are radial and named radial Bessel-sinc (RBS) functions being time-frequency atoms with nonlinear phase. We obtain a recursive formula for the RBS functions in R d with d being odd. Based on the RBS function, a corresponding sampling theorem for a class of non-bandlimited signals is established. We investigate a class of radial functions and prove that each of these functions can be extended to become a monogenic function between two parallel planes, where the monogencity is taken to be of the Clifford analysis sense.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

UDC 517.5 A useful tool proved in the present paper is the Borel–Pompeiu formula for the resulting function class of null solutions of the quaternionic perturbed fractional $\psi$-Fueter operator. Motivated by this formula, we developed major steps of the associated Bergman theory.