scholarly journals A local representation formula for quaternionic slice regular functions

2020 ◽  
pp. 1
Author(s):  
Graziano Gentili ◽  
Caterina Stoppato
Keyword(s):  
Representation Formula ◽  
Slice Regular Functions ◽  
Local Representation ◽  
Regular Functions

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 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.

scholarly journals A Phragmén - Lindelöf principle for slice regular functions

2011 ◽  
Vol 18 (4) ◽  
pp. 749-759 ◽  
Author(s):  
Graziano Gentili ◽  
Caterina Stoppato ◽  
Daniele C. Struppa
Keyword(s):  
Slice Regular Functions ◽  
Regular Functions ◽  
Lindelöf Principle

scholarly journals Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

2020 ◽  
Vol 30 (4) ◽  
Author(s):  
Alessandro Perotti
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Zonal Harmonic ◽  
Slice Regular Functions ◽  
Regular Functions ◽  
Quaternionic Polynomials ◽  
Mean Value Formula

Abstract We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ h 1 , $$h_2$$ h 2 of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ f = h 1 - x ¯ h 2 . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

On Two Approaches to the Bergman Theory for Slice Regular Functions

2013 ◽  
pp. 39-54 ◽  
Author(s):  
Fabrizio Colombo ◽  
J. Oscar González-Cervantes ◽  
Maria Elena Luna-Elizarrarás ◽  
Irene Sabadini ◽  
Michael Shapiro
Keyword(s):  
Slice Regular Functions ◽  
Regular Functions
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