Slice Regular Functions Research Articles

On the geometry of zero sets of central quaternionic polynomials

Let R be the ring H[x1,…,xn] of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if p∈R vanishes at all the common zeros of I in Hn with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hn. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

An approach to slice regular functions via post‐quantum calculus theory

The purpose of this paper is to combine the ‐calculus in the quaternionic context, which is proposed via two kinds of ‐operators, with the theory of slice regular functions. Specifically, we shall work in suitable subclasses of slice regular functions in which the ‐operators can be related with the slice derivative. The paper presents some results such as an integral formula and series expansion.

A unified notion of regularity in one hypercomplex variable

We define a very general notion of regularity for functions taking values in an alternative real ⁎-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail.

Almansi-Type Decomposition for Slice Regular Functions of Several Quaternionic Variables

AbstractIn this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields $$2^n$$ 2 n distinct and unique decompositions for any slice function with domain in $$\mathbb {H}^n$$ H n . Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter’s Theorem in $$\mathbb {H}^n$$ H n , establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.

Quaternionic slice regularity beyond slice domains

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.

An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory

An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory

Partial slice regularity and Fueter's theorem in several quaternionic variables

Abstract We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions that are slice, slice regular, and circular with respect to given variables are characterized. We introduce new notions of partial spherical value and derivative for functions of several variables that extend those of one variable. We recover some of their properties as circularity, harmonicity, some relations with differential operators, and a Leibniz rule with respect to the slice product as well as studying their behavior in the context of several variables. Then, we prove our main result, which is a generalization of Fueter’s theorem for slice regular functions in several variables. This extends the link between slice regular and axially monogenic functions well known in the one variable context.

Fractional Slice Regular Functions of a Quaternionic Variable

Fractional Slice Regular Functions of a Quaternionic Variable

$*$-logarithm for slice regular functions

In this paper, we study the (possible) solutions of the equation \exp_*(f)=g , where g is a slice regular never vanishing function on a circular domain of the quaternions \mathbb{H} and \exp_* is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function f which satisfies \exp_*(f)=g is called a * -logarithm of g . We provide necessary and sufficient conditions, expressed in terms of the zero set of the “vector” part g_v of g , for the existence of a * -logarithm of g , under a natural topological condition on the domain \Omega . By this way, we prove an existence result if g_v has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of g_v are finite, the domain is either the unit ball, or \mathbb{H} , or \mathbb{D} (the solid torus obtained by circularization in \mathbb{H} of the disc contained in \mathbb{C} and centered in 2\sqrt{-1} with radius 1 ), and a further condition on the “real part” g_0 of g is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of g_v , in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.

On a definition of logarithm of quaternionic functions

For a slice-regular quaternionic function f , the classical exponential function \exp f is not slice-regular in general. An alternative definition of an exponential function, the * -exponential \exp_* , was given in the work by Altavilla and de Fabritiis (2019): if f is a slice-regular function, then \exp_*f is a slice-regular function as well. The study of a * -logarithm \log_*f of a slice-regular function f becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a \log_*f depends only on the structure of the zero set of the vectorial part f_v of the slice-regular function f=f_0+f_v , besides the topology of its domain of definition. We also show that, locally, every slice-regular nonvanishing function has a * -logarithm and, at the end, we present an example of a nonvanishing slice-regular function on a ball which does not admit a * -logarithm on that ball.

A Local Cauchy Integral Formula for Slice-Regular Functions

AbstractWe prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.

A Note on Location of the Zeros of Quaternionic Polynomials

The purpose of this paper is to investigate the extensions of the classical Eneström-Kakeya theorem and its various generalizations concerning the distribution of zeros of polynomials from the complex to the quaternionic setting. Using a maximum modulus theorem and the zero set structure in the recently published theory of regular functions and polynomials of a quaternionic variable, we construct new bounds of the Eneström-Kakeya type for the zeros of these polynomials. The obtained results for this subclass of polynomials and slice regular functions give generalizations of a number of results previously reported in the relevant literature.

A representation formula for slice regular functions over slice-cones in several variables

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form {mathbb {R}}^{2n}. This is a new setting which contains and encompasses in a nontrivial way other cases already studied in the literature and which requires new tools. To this end, we define a cone {mathcal {W}}_{mathcal {C}}^d in [{text {End}}({mathbb {R}}^{2n})]^d and we extend the slice topology tau _s to this cone. Slice regular functions can be defined on open sets in left( tau _s,{mathcal {W}}_{mathcal {C}}^dright) and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative *-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

Adaptive Fourier Decomposition of Slice Regular Functions

In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator \(\mathcal S_a\) with the decomposition process $$\begin{aligned} f=e_a\langle f, e_a\rangle +B_{a}*\mathcal S_a f, \end{aligned}$$where \(e_a\) denotes the slice normalized Szegö kernel and \( B_a \) the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka–Malmquist orthonormal system.

Quaternionic Slice Regular Functions and Quaternionic Laplace Transforms

The functions studied in the paper are the quaternion-valued functions of a quaternionic variable. It is shown that the left slice regular functions and right slice regular functions are related by a particular involution, and that the intrinsic slice regular functions play a central role in the theory of slice regular functions. The relation between left slice regular functions, right slice regular functions and intrinsic slice regular functions is revealed. As an application, the classical Laplace transform is generalized naturally to quaternions in two different ways, which transform a quaternion-valued function of a real variable to a left or right slice regular function. The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.

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

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

Extension theorem and representation formula in non-axially-symmetric domains for slice regular functions

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of f(q)=\sum_{n\in\mathbb{N}}(q-p)^{*n} a_n , given by a \sigma -ball \Sigma(p,r) , is not open in \mathbb{H} unless p\in\mathbb{R} . This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along these lines, we can establish an extension theorem and a representation formula in a slice domain.

Slice regular functions and orthogonal complex structures over $\mathbb{R}^8$

This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the open mapping theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions.

Slice conformality and Riemann manifolds on quaternions and octonions

In this paper we establish quaternionic and octonionic analogs of the classical Riemann surfaces. The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal or slice isothermal parameterization of quaternionic or octonionic Riemann manifolds. These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs of slice regular functions, the 4 and 8 dimensional spheres, the helicoidal and catenoidal 4 and 8 dimensional manifolds. Using appropriate Riemann manifolds, we also give a unified definition of the quaternionic and octonionic logarithm and n-th root function.

Some characterizations of slice regular Lipschitz type spaces

We give some characterizations of Lipschitz type spaces of slice regular functions in the unit ball of the skew field of quaternions with prescribed modulus of continuity.