scholarly journals A New Approach to Slice Analysis Via Slice Topology

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Xinyuan Dou ◽  
Ming Jin ◽  
Guangbin Ren ◽  
Irene Sabadini
Keyword(s):  
Taylor Expansion ◽  
Regular Function ◽  
Alternative Algebra ◽  
Representation Formula ◽  
Monogenic Functions ◽  
Slice Regular Functions ◽  
New Approach ◽  
Regular Functions ◽  
Several Variables ◽  
Slice Monogenic Functions

AbstractIn this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.

scholarly journals The Harmonicity of Slice Regular Functions

2020 ◽  
Author(s):  
Cinzia Bisi ◽  
Jörg Winkelmann
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Regular Function ◽  
Representation Formula ◽  
Mean Value ◽  
Quaternionic Analysis ◽  
Poisson Formula ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Mean Value Formula

Abstract In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over $${\mathbb {H}}$$ H . Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over $${\mathbb {H}}$$ H (analogous to an holomorphic function over $${\mathbb {C}}$$ C ) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over $${\mathbb {H}}$$ H ? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over $${\mathbb {H}}$$ H and a Jensen’s Formula for semi-regular ones.

scholarly journals Spherical Coefficients of Slice Regular Functions

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Amedeo Altavilla
Keyword(s):  
Differential Equations ◽  
Regular Function ◽  
Countable Family ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Spherical Expansion ◽  
Derivatives Of

AbstractGiven a quaternionic slice regular function f, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of f with those of its slice derivative $$\partial _{c}f$$ ∂ c f obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.

scholarly journals Landau’s theorem for slice regular functions on the quaternionic unit ball

2017 ◽  
Vol 28 (03) ◽  
pp. 1750017 ◽  
Author(s):  
Cinzia Bisi ◽  
Caterina Stoppato
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Analogous Result ◽  
Open Unit ◽  
Open Unit Ball ◽  
Slice Regular Functions ◽  
New Approach ◽  
The Real ◽  
Regular Functions ◽  
Real Algebra

During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].

scholarly journals On the real differential of a slice regular function

2018 ◽  
Vol 18 (1) ◽  
pp. 5-26 ◽  
Author(s):  
Amedeo Altavilla
Keyword(s):  
General Result ◽  
Regular Function ◽  
Singular Set ◽  
Slice Regular Functions ◽  
The Real ◽  
Regular Functions ◽  
New Information ◽  
Spherical Expansion ◽  
New Formula ◽  
Derivatives Of

AbstractIn this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (calledspherical expansion), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.

scholarly journals Weak Slice Regular Functions on the n-Dimensional Quadratic Cone of Octonions

2021 ◽  
Author(s):  
Xinyuan Dou ◽  
Guangbin Ren ◽  
Irene Sabadini ◽  
Ting Yang
Keyword(s):  
Complex Plane ◽  
Taylor Expansion ◽  
Complex Structure ◽  
Quadratic Cone ◽  
Axially Symmetric ◽  
Slice Regular Functions ◽  
New Class ◽  
Regular Functions ◽  
The One ◽  
Class Of Functions

AbstractIn the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called n-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion.

scholarly journals Further properties of the Bergman spaces of slice regular functions

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Fabrizio Colombo ◽  
J. Oscar González-Cervantes ◽  
Irene Sabadini
Keyword(s):  
Unit Ball ◽  
Half Space ◽  
Half Plane ◽  
Bergman Kernel ◽  
Bergman Spaces ◽  
Reflection Principle ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Complex Half Plane ◽  
Schwarz Reflection Principle

AbstractWe continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

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