Axial Monogenic Functions from Holomorphic Functions

Common, A. K.; Sommen, Franciscus

doi:10.1006/jmaa.1993.1372

Cited by 19 publications

(14 citation statements)

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“…We have proved that the functionf (x 0 , ρ) is axially monogenic (see [12]) and that ∆ (n−1)/2 is surjective onto AM(U ). Moreover, givenf ∈ AM(U ) we construct a slice monogenic function f satisfying (1), thus inverting the Fueter mapping theorem.…”

Section: Introduction and Notationsmentioning

confidence: 95%

The inverse Fueter mapping theorem in integral form using spherical monogenics

2012

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In this paper we prove an integral representation formula for the inverse Fueter mapping theorem for monogenic functions defined on axially symmetric open sets U ⊆ R n+1 , i.e. on open sets U invariant under the action of SO(n). Every monogenic function on such an open set U can be written as a series of axially monogenic functions of degree k, i.e. functions of typef k (x) := [A(x 0 , ρ) + ωB(x 0 , ρ)]P k (ω), where A(x 0 , ρ) and B(x 0 , ρ) satisfy a suitable Vekua-type system and P k (ω) are spherical monogenic polynomials of degree k. The Fueter mapping theorem says that given a holomorphic function f of a paravector variable defined on U then the functionf (x)P k (x) given byis a monogenic function. The aim of this paper is to invert the Fueter mapping theorem by determining a holomorphic function f of a paravector variable in terms off (x)P k (x). This result allows to invert the Fueter mapping theorem for any monogenic function defined on an axially symmetric open set.

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Section: Introduction and Notationsmentioning

confidence: 95%

The inverse Fueter mapping theorem in integral form using spherical monogenics

2012

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“…Let us start with the following lemma, which links Riemann-Hilbert problems for axially monogenic functions on upper half unit ball of R 4 with Riemann-Hilbert problems for analytic functions over upper half disc of the complex plane. More details about the construction principle can be found in [10,11,15,20,21,31].…”

Section: Riemann-hilbert Problems For Axially Monogenic Functionsmentioning

confidence: 99%

“…where g 2 is given by (12), and Π is given by the Riemann-Hilbert problem (8). Therefore, similar to (11), one has…”

Section: Riemann-hilbert Problems For Axially Monogenic Functionsmentioning

confidence: 99%

“…The basic method is to apply Fueter's theorem, see e.g. [10,11,15,31] for the four dimensional case, and to transfer the Riemann-Hilbert problem to an equivalent Riemann-Hilbert problem for analytic functions over the complex plane. However, so far, there are no results about Riemann-Hilbert problems with variable coefficients for axially monogenic functions defined over special domains in four dimensions, like the upper half unit ball, upper half space, and the quadrant.…”

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confidence: 99%

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Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of $${\mathbb {R}}^4$$ R 4

Ku¹,

Wang²,

He³

et al. 2017

Adv. Appl. Clifford Algebras

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In this paper we are interested in finding solutions to Riemann-Hilbert boundary value problems, for short Riemann-Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann-Hilbert problems for axially monogenic functions defined over the upper half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann-Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy-Riemann equations. Mathematics SubjectClassification. Primary: 30E25, 35Q15, 31A25, 31B20; Secondary: 31B10, 35J56, 35J58.2494 M. Ku et al.Adv. Appl. Clifford Algebras boundary value problems, their study is closely connected with the theory of singular integral equations [17,30] and has a wide range of applications in other fields, such as in the theory of cracks and elasticity [22,30], in quantum mechanics and of statistical physics [9] as well as in the theory of linear and nonlinear partial differential equations [14], in the theory of orthogonal polynomials and asymptotic analysis [12], and in the theory of time-frequency analysis [1]. In addition, their methods and related problems have been extended to null-solutions to complex partial differential equations (PDEs) on the complex plane like poly-analytic functions, meta-analytic functions, and poly-harmonic functions, see, e.g. [5]. Moreover, also recently, the boundary value theory of null-solutions to complex PDEs has been explored on special domains of the complex plane, like the unit disk, upper half plane, half disks, circular rings, triangles, and sectors, and so on, see, e.g. [5,6,36].In parallel, there have been many attempts of generalizing the classical boundary value theory of Riemann-Hilbert problems into higher dimensions, mainly by considering two principal ways, i.e., the theory of several complex variables and Clifford analysis, in particular quaternionic analysis. The latter is an elegant generalization of the classical theory of complex analysis. It concentrates on the study of the theory of so-called monogenic functions, and refines real harmonic analysis in the sense that its principal operator, the Dirac or generalized Cauchy-Riemann operator, factorizes the higher-dimensional Laplace operator, see, e.g. [7,13,18]. In the setting of quaternionic and Clifford analysis, Riemann-Hilbert problems were discussed by many authors, see, e.g. ], the authors studied Riemann-Hilbert problems with constant coefficients. Their solutions were given explicitly in terms of Cauchy-type integral operators together with power series expansions. Moreover, these problems are closely connected with applications, like the theory of fluid mechanics [18] and signal processing in higher dimensions [16,32]. To the authors' knowledge in Refs. [20,21] the authors made the first ...

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“…where C q p,j , A p,q and B p,q are real-valued, and x → A p,q (x 0 , |x|)+xB p,q (x 0 , |x|) is an axial monogenic function ( [3,10]) when q = 0. From Proposition 3.1 (see the next section) we conclude that…”

Section: Preliminariesmentioning

confidence: 99%

Holomorphic approximation ofL2-functions on the unit sphere inR3

Schepper

Qian

Sommen

et al. 2014

Journal of Mathematical Analysis and Applications

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Abstract:In this paper we construct an embedding of holomorphic functions in two complex variables into the unit ball in R^3. This leads to a closed subspace of the L2-functions on the unit sphere spanned by quaternionic polynomials for which we construct orthonormal bases and study the related convergence properties. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems CorporationHolomorphic approximation of L 2 -functions on the unit sphere in R 3Nele De Schepper, Tao Qian, Frank Sommen and Jinxun WangAbstract. In this paper we construct an embedding of holomorphic functions in two complex variables into the unit ball in R 3 . This leads to a closed subspace of the L2-functions on the unit sphere spanned by quaternionic polynomials for which we construct orthonormal bases and study the related convergence properties.Mathematics Subject Classification (2010). 33C47.

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Axial Monogenic Functions from Holomorphic Functions

Cited by 19 publications

References 0 publications

The inverse Fueter mapping theorem in integral form using spherical monogenics

The inverse Fueter mapping theorem in integral form using spherical monogenics

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of $${\mathbb {R}}^4$$ R 4

Holomorphic approximation ofL2-functions on the unit sphere inR3

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