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# Fueter’s Theorem for Monogenic Functions in Biaxial Symmetric Domains

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• Qrama - Ghent University
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## Abstract

Fueter’s theorem discloses a remarkable connection existing between holomorphic functions and monogenic functions in $$\mathbb R^{m+1}$$ when m is odd. It states that $$\Delta _{m+1}^{k+\frac{m-1}{2}}\bigl [\bigl (u(x_0,\vert \underline{x}\vert )+\frac{\underline{x}}{\vert \underline{x}\vert }\,v(x_0,\vert \underline{x}\vert )\bigr )P_k(\underline{x})\bigr ]$$ is monogenic if $$u+iv$$ is holomorphic and $$P_k(\underline{x})$$ is a homogeneous monogenic polynomial in $$\mathbb R^m$$. Eelbode et al. (AIP Conf Proc 1479:340–343, 2012) proved that this statement is still valid if the monogenicity condition on $$P_k(\underline{x})$$ is dropped. To obtain this result, the authors used representation theory methods but their result also follows from a direct calculus we established in our paper Peña Peña and Sommen (J Math Anal Appl 365:29–35, 2010). In this paper we generalize the result from Eelbode et al. (2012) to the case of monogenic functions in biaxially symmetric domains. In order to achieve this goal we first generalize Peña Peña and Sommen (2010) to the biaxial case and then derive the main result from that.
Results Math 72 (2017), 1747–1758
c
2017 Springer International Publishing AG
1422-6383/17/041747-12
published online August 8, 2017
DOI 10.1007/s00025-017-0732-2 Results in Mathematics
Fueter’s Theorem for Monogenic Functions
in Biaxial Symmetric Domains
Dixan Pe˜na Pe˜na, Irene Sabadini, and Franciscus Sommen
Abstract. Fueter’s theorem discloses a remarkable connection existing be-
tween holomorphic functions and monogenic functions in Rm+1 when m
is odd. It states that Δk+m1
2
m+1 u(x0,|x|)+ x
|x|v(x0,|x|)Pk(x)is mono-
genic if u+iv is holomorphic and Pk(x) is a homogeneous monogenic
polynomial in Rm. Eelbode et al. (AIP Conf Proc 1479:340–343, 2012)
proved that this statement is still valid if the monogenicity condition on
Pk(x) is dropped. To obtain this result, the authors used representation
theory methods but their result also follows from a direct calculus we
established in our paper Pe˜na Pe˜na and Sommen (J Math Anal Appl
365:29–35, 2010). In this paper we generalize the result from Eelbode
et al. (2012) to the case of monogenic functions in biaxially symmetric
domains. In order to achieve this goal we ﬁrst generalize Pe˜na Pe˜na and
Sommen (2010) to the biaxial case and then derive the main result from
that.
Mathematics Subject Classiﬁcation. 30G35, 31A05.
Keywords. Cliﬀord monogenic functions, Fueter’s theorem, Fischer
decomposition.
1. Introduction
Let Rmbe the real Cliﬀord algebra generated by the standard basis {e1,...,e
m}
of the Euclidean space Rm(see [2,17]). The multiplication in this associative al-
gebra is determined by the relations: e2
j=1, ejek+ekej=0,1j=km.
Any Cliﬀord number aRmmay thus be written as
a=
A
aAeA,a
AR,
where the basis elements eA=ej1...e
jkare deﬁned for every subset A=
{j1,...,j
k}of {1,...,m}with j1<···<j
k(for A=one puts e= 1).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... For the theory of axially monogenic functions of degree zero and also of positive degree, we refer the reader to [8,37,46,47] and [10, Chapter II, Section 2.4]. For recent developments on this topic, see [5,12,39] and their references. ...
... Repeating the preceding argument, we obtain that the same equality holds for each x ∈ O \ cl(Ω D ). This proves (39). ...
Preprint
Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:\Omega_D\subset\mathbb{O}\to\mathbb{O}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $\mathbb{H}_\mathbb{I}$ of $\mathbb{O}$ generated by a pair $\mathbb{I}=(I,J)$ of orthogonal imaginary units $I$ and $J$ ($\mathbb{H}_\mathbb{I}$ is a quaternionic slice' of $\mathbb{O}$), the restriction of $f$ to $\Omega_D\cap\mathbb{H}_\mathbb{I}$ belongs to the kernel of the corresponding Cauchy-Riemann-Fueter operator $\frac{\partial}{\partial x_0}+I\frac{\partial}{\partial x_1}+J\frac{\partial}{\partial x_2}+(IJ)\frac{\partial}{\partial x_3}$. The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their holomorphic nature': slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine $8$-dimesional domains of $\mathbb{O}$. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator $\Gamma$ and of slice Fueter operator $\bar{\vartheta}_F$ over octonions, which allow to characterize slice Fueter-regular functions as the $\mathscr{C}^2$-functions in the kernel of $\bar{\vartheta}_F$ satisfying a second order differential system associated with $\Gamma$. The paper contains eight open problems.
... The FSQ-mapping theorem and its generalizations can be found in [27,28,29,36,41,42,48], more recently there has been an intensive research in the direction of the inverse FSQ-mapping theorem, which has been investigated in the papers [9,12,13,14,25,26]. Using the Radon and dual Radon transform, see [6], it is possible to find a different method, with respect to the FSQtheorem, to relate slice monogenic functions and the monogenic functions. ...
Preprint
Full-text available
The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure to extend holomorphic functions of one complex variable to slice monogenic functions and to monogenic functions. Using the Cauchy formula of slice monogenic functions the FSQ-theorem admits an integral representation for $n$ odd. In this paper we show that the relation $\Delta_{n+1}^{(n-1)/2}S_L^{-1}=\mathcal{F}^L_n$ between the slice monogenic Cauchy kernel $S_L^{-1}$ and the F-kernel $\mathcal{F}^L_n$, that appear in the integral form of the FSQ-theorem for $n$ odd, holds also in the case we consider the fractional powers of the Laplace operator $\Delta_{n+1}$ in dimension $n+1$, i.e., for $n$ even. Moreover, this relation is proven computing explicitly Fourier transform of the kernels $S_L^{-1}$ and $\mathcal{F}^L_n$ as functions of the Poisson kernel. Similar results hold for the right kernels $S_R^{-1}$ and of $\mathcal{F}^R_n$.
... Observe that for the Fueter's theorem the operator T F S2 is equal to the Laplacian ∆ in 4 dimensions. Further developments can be found in [71,72,73] see also the survey [76]. ...
Preprint
Full-text available
The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collaborators. One of the main goals of this paper is to show that the $\mathcal{F}$-functional calculus generates the Riesz projectors. The existence of such projectors is obtained via the $\mathcal{F}$-resolvent equation that we have generalized to the Clifford algebra setting. This equation was known in the quaternionic setting, but the Clifford algebras setting turned out to be much more complicated.
... Observe that for the Fueter's theorem the operator T F S2 is equal to the Laplacian Δ in 4 dimensions. Further developments can be found in [75][76][77] see also the survey [80]. ...
Article
Full-text available
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n -tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S -spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F -functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S -spectrum.
... Observe that for the Fueter's theorem the operator T F S2 is equal to the laplacian ∆ in 4 dimensions. Further developments can be found in [71,72,73] see also the survey [76]. We can summarize the Fueter-Sce contractions as follows. ...
Preprint
Full-text available
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for $n$-tuples of operators $(A_1,...,A_n)$. A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the $S$-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the $F$-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. Here we also discuss how to define the fractional Fourier's law for nonhomogeneous materials, such definition is based on the spectral theory on the $S$-spectrum.
Article
The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure to extend holomorphic functions of one complex variable to slice monogenic functions and to monogenic functions. Using the Cauchy formula of slice monogenic functions the FSQ-theorem admits an integral representation for n odd. In this paper we show that the relation Δn+1(n−1)/2SL−1=FnL between the slice monogenic Cauchy kernel SL−1 and the F-kernel FnL, that appear in the integral form of the FSQ-theorem for n odd, holds also in the case we consider the fractional powers of the Laplace operator Δn+1 in dimension n+1, i.e., for n even. Moreover, this relation is proven computing explicitly the Fourier transform of the kernels SL−1 and FnL as functions of the Poisson kernel. Similar results hold for the right kernels SR−1 and of FnR.
Article
Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra $${\mathbb {O}}$$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function $$f:\Omega _D\subset {\mathbb {O}}\rightarrow {\mathbb {O}}$$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $${\mathbb {H}}_{\mathbb {I}}$$ of $${\mathbb {O}}$$ generated by a pair $${\mathbb {I}}=(I,J)$$ of orthogonal imaginary units I and J ($${\mathbb {H}}_{\mathbb {I}}$$ is a ‘quaternionic slice’ of $${\mathbb {O}}$$), the restriction of f to $${\mathbb {H}}_{\mathbb {I}}$$ belongs to the kernel of the corresponding Cauchy–Riemann–Fueter operator $$\frac{\partial }{\partial x_0}+I\frac{\partial }{\partial x_1}+J\frac{\partial }{\partial x_2}+(IJ)\frac{\partial }{\partial x_3}$$. The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their ‘holomorphic nature’: slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine 8-dimesional domains of $${\mathbb {O}}$$. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator $$\Gamma$$ and of slice Fueter operator $${\overline{\vartheta }}_F$$ over octonions, which allow to characterize the slice Fueter-regular functions as the $${\mathscr {C}}^2$$-functions in the kernel of $${\overline{\vartheta }}_F$$ satisfying a second order differential system associated with $$\Gamma$$.
Chapter
M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., (8) 23 (1957), 220–225 as well as some comments and historical remarks.
Chapter
Fueter’s Theorem offers a method that conveys certain holomorphic functions in one complex variable to quaternionic regular (monogenic) functions. Ever since the theorem was proved in 1935, it underwent several main generalizations. Those are not only based on its own interest, but also motivated by applications found in other areas of mathematics, such as functional calculus of operators. This article serves as a survey on Fueter’s Theorem, its generalizations and applications.
Article
In this paper we discuss a generalization of Fueter's theorem which states that whenever f(x0,x) is holomorphic in x0 + x then it satisfies D□f = 0, D = ∂x0 + i∂x1 + j∂x2 + kdx3 being the Fueter operator.
Article
In the recent papers [Commun. Pure Appl. Anal. 10, No. 4, 1165–1181 (2011; Zbl 1258.30022); Isr. J. Math. 194, Part A, 485–505 (2013; Zbl 06176555)], we have started a systematic study of the inverse Fueter mapping theorem. In this paper, we show that the inversion theorem holds for the case of biaxially monogenic functions. Here there are several additional difficulties with respect to the cases already treated. However, we are still able to prove an integral version of the inverse Fueter mapping theorem. The kernels appearing in the integral representation formula have an explicit representation that can be computed depending on the dimension of the Euclidean space in which the problem is considered.
Book
* Preface * Background Material * Computational Algebraic Analysis for Systems of Linear Constant Coefficients Differential Equations * The Cauchy-Fueter Systems and its Variations * Special First Order Systems in Clifford Analysis * Some First Order Linear Operators in Physics * Open Problems and Avenues for Further Research * References * Index
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A historical review of spinors is given together with a construction of spinor spaces as minimal left ideals of Clifford algebras. Spinor spaces of euclidean spaces over reals have a natural linear structure over reals, complex numbers or quaternions. Clifford algebras have involutions which induce bilinear forms or scalar products on spinor spaces. The automorphism groups of these scalar products of spinors are determined and also classified.