The inverse Fueter mapping theorem in integral form using spherical monogenics

Colombo, Fabrizio; Sabadini, Irene; Sommen, Franciscus

doi:10.1007/s11856-012-0090-4

Cited by 25 publications

(16 citation statements)

References 23 publications

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“…Both these theories are natural generalizations of the complex function theory but they are quite different from each other. Possible relations between the two theories were developed in odd dimensions in the context of the Fueter construction (which allows to construct monogenic functions starting from holomorphic functions) and its inversion ( [3,4]). The possibility to relate the two theories in even dimensions, using the Fueter mapping technique is still under investigation.…”

Section: Introductionmentioning

confidence: 99%

The Radon transform between monogenic and generalized slice monogenic functions

et al. 2015

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In [2], the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on 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 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 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.

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Section: Introductionmentioning

confidence: 99%

The Radon transform between monogenic and generalized slice monogenic functions

et al. 2015

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“…which corresponds to (16) for h = 1. Let us assume that the formula (16) holds for some h ∈ N and let us show that it holds for h + 1.…”

Section: The Fueter-sce Mapping Theorem In Integral Formmentioning

confidence: 99%

Formulations of the -functional calculus and some consequences

Colombo¹,

Gantner²

2016

Proceedings of the Royal Society of Edinburgh: Section A Mathem
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In this paper we introduce the two possible formulations of the F-functional calculus which are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo F-resolvent equation. In the case of dimension 3 we prove the F-resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the F-functional calculus.AMS Classification: 47A10, 47A60. Key words: Fueter-Sce mapping theorem in integral form, the F-spectrum, formulations of the Ffunctional calculus for n-tuples of operators, projectors, quaternionic version of the F -functional calculus, quaternionic F-resolvent equation.

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“…The literature on hyperholomorphic function theories and related spectral theories is nowadays very large. For the function theory of slice hyperholomorphic functions the main books are [6,46,48,56,60], while for the spectral theory on the S-spectrum we mention the books [5,23,24,48]. For the Fueter and monogenic function theory and related topics see the books [15,44,52,65,66,73,82].…”

Section: Introductionmentioning
confidence: 99%

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators
Colombo
¹
,
Gantner
²
,
Pinton
³

2021
Adv. Appl. Clifford Algebras
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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.

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The inverse Fueter mapping theorem in integral form using spherical monogenics

Cited by 25 publications

References 23 publications

The Radon transform between monogenic and generalized slice monogenic functions

The Radon transform between monogenic and generalized slice monogenic functions

Formulations of the -functional calculus and some consequences

An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

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