F. Sommen

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Clifford Algebra and Spinor-Valued Functions

1992-01-01

Richard Delanghe , F. Sommen , Vladimı́r Souček

Clifford algebra and spinor-valued functions (a function theory for the Dirac operator)

1992-11-01

Richard Delanghe , F. Sommen , Vladimı́r Souček

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Analysis of Dirac Systems and Computational Algebra

2004-01-01

Fabrizio Colombo , Irene Sabadini , F. Sommen +1 more

* 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 … * 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

Lie groups as spin groups

1993-08-01

C. J. L. Doran , David Hestenes , F. Sommen +1 more

It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of … It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.

Fundaments of Hermitean Clifford analysis part II: splitting ofh-monogenic equations

2007-10-01

Fred Brackx , Jarolím Bureš , Hennie De Schepper +3 more

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and … Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2.

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Spherical harmonics and integration in superspace: II

2009-05-28

Hendrik De Bie , David Eelbode , F. Sommen

The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007 The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007

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On a Generalization of Fueter’s Theorem

2000-12-31

F. Sommen

In this paper we discuss a generalization of Fueter’s theorem which states that whenever f(x_0, x) is holomorphic in x_0+x , then it satisfies D?f = 0, D= \partial_{x_0} + … In this paper we discuss a generalization of Fueter’s theorem which states that whenever f(x_0, x) is holomorphic in x_0+x , then it satisfies D?f = 0, D= \partial_{x_0} + i \partial_{x_1}+ j \partial_{x_2} + k \partial_{x_3} being the Fueter operator.

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Rarita–Schwinger Type Operators in Clifford Analysis

2001-10-01

Jarolím Bureš , F. Sommen , Vladimı́r Souček +1 more

Spherical harmonics and integration in superspace

2007-06-12

Hendrik De Bie , F. Sommen

In this paper, the classical theory of spherical harmonics in R m is extended to superspace using techniques from Clifford analysis.After defining a super-Laplace operator and studying some basic properties … In this paper, the classical theory of spherical harmonics in R m is extended to superspace using techniques from Clifford analysis.After defining a super-Laplace operator and studying some basic properties of polynomial nullsolutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti.This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace.Finally, this integration over the supersphere is used to define an integral over the whole superspace, and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.

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Generalizations of Fueter’s theorem

2002-01-01

Kit Ian Kou , Tao Qian , F. Sommen

In relation to the solution of the Vekua system for axial type monogenic functions, generalizations of Fueter's Theorem are discussed.We show that if f is a holomorphic function in one … In relation to the solution of the Vekua system for axial type monogenic functions, generalizations of Fueter's Theorem are discussed.We show that if f is a holomorphic function in one complex variable, then for any underlying space R n 1 the induced function ∆ k+(n-1)/2 f (x 0 +x)P k (x), where P k (x) is left-monogenic and homogeneous of degree k, is left-monogenic whenever k +(n-1)/2 is a non-negative integer.If the space dimension n + 1 is odd, then the above also holds for k being non-negative integers.

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Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains

2005-01-01

Paula Cerejeiras , Uwe Kähler , F. Sommen

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier–Stokes equations over time-varying domains. Copyright © 2005 John … We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier–Stokes equations over time-varying domains. Copyright © 2005 John Wiley & Sons, Ltd.

Models for irreducible representations ofSpin(m)

2001-02-01

P. Van Lancker , F. Sommen , Denis Constales

Some connections between clifford analysis and complex analysis

1982-09-01

F. Sommen

In the paper we construct kernels and mono-genic and holomorphic in in order to extend to several dimersions respectively the tranformation given by and the Fourier-Borel transformation T belonging to … In the paper we construct kernels and mono-genic and holomorphic in in order to extend to several dimersions respectively the tranformation given by and the Fourier-Borel transformation T belonging to a space of analytic functionals. This leads toconnections between the theory of holomorphic functions of several variables and the theory of monogenic functions. These relationships are used to study the absolute convergence of the multiple Taylor series for monogenic functions.

On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

2009-09-01

Fred Brackx , Bram De Knock , Hennie De Schepper +1 more

On conjugate harmonic functions in Euclidean space

2002-11-10

Fred Brackx , Richard Delanghe , F. Sommen

Abstract In this paper we consider the problem of constructing in domains Ωof ℝ m +1 with a specific geometric property, a conjugate harmonic V to a given harmonic function … Abstract In this paper we consider the problem of constructing in domains Ωof ℝ m +1 with a specific geometric property, a conjugate harmonic V to a given harmonic function U , as a direct generalization of the complex plane case. This construction is carried out in the framework of Clifford analysis which focusses on the so‐called monogenic functions, i.e. null solutions of the Dirac operator. An explicit formula of the associated monogenic function F = U + ē 0 V in terms of a harmonic potential is constructed and the interconnection with the Stein–Weiss notion of conjugate harmonicity will be shown. Copyright © 2002 John Wiley & Sons, Ltd.

Deriving Harmonic Functions in Higher Dimensional Spaces

2003-06-30

Tao Qian , F. Sommen

For a harmonic function, by replacing its variables with norms of vectors in some multi-dimensional spaces, we may induce a new function in a higher dimensional space. We show that, … For a harmonic function, by replacing its variables with norms of vectors in some multi-dimensional spaces, we may induce a new function in a higher dimensional space. We show that, after applying to it a certain power of the Laplacian, we obtain a new harmonic function in the higher dimensional space. We show that Poisson and Cauchy kernels and Newton potentials, and even heat kernels are all deducible using this method based on their forms in the lowest dimensional spaces. Fueter's theorem and its generalizations are deducible as well from our results. The latter has been used to singular integral and Fourier multiplier theory on the unit spheres and their Lipschitz perturbations of higher dimensional Euclidean spaces.

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Clifford-hermite wavelets in euclidean space

2000-05-01

Fred Brackx , F. Sommen

The Fourier transform in Clifford analysis

2009-01-01

Fred Brackx , Nele De Schepper , F. Sommen

Hypercomplex Fourier and Laplace transforms I

1982-06-01

F. Sommen

In this paper the exponential function introduced in [14] and the Cauchy kernel introduced in [7] are used to study new kinds of Fourier, Laplace, Cauchy and Hilbert transforms of … In this paper the exponential function introduced in [14] and the Cauchy kernel introduced in [7] are used to study new kinds of Fourier, Laplace, Cauchy and Hilbert transforms of A valued L 2-func...

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AN ALGEBRA OF ABSTRACT VECTOR VARIABLES

1997-01-01

F. Sommen

In this paper we introduce an abstract algebra of vector variables that generalizes both polynomial algebra and Clifiord algebra. This abstractly deflned algebra and its endomorphisms contains all the basic … In this paper we introduce an abstract algebra of vector variables that generalizes both polynomial algebra and Clifiord algebra. This abstractly deflned algebra and its endomorphisms contains all the basic SO(m)-invariant polynomials and operators used in Clifiord analysis.

The generalized clifford-hermite continuous wavelet transform

2001-02-01

Fred Brackx , F. Sommen

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Monogenic Differential Operators

1992-11-01

F. Sommen , N. Van Acker

In this paper we consider operators acting on Clifford algebra valued polynomials and, in particular, differential operators with polynomial coefficients. The decomposition of polynomials into homogeneous pieces leads to the … In this paper we consider operators acting on Clifford algebra valued polynomials and, in particular, differential operators with polynomial coefficients. The decomposition of polynomials into homogeneous pieces leads to the classical homogeneous decomposition of operators and the further decomposition of homogeneous polynomials into monogenic polynomials leads to the concept of monogenic operator. Monogenic operators are characterized in terms of commutation relations and the monogenic decomposition of differential operators is studied in detail.

Martinelli–Bochner Type Formulae in Complex Clifford Analysis

1987-02-28

F. Sommen

Various types of solutions of the systems (D_x + iD_y)f = 0 are considered, where D_x and D_y are Dirac type operators in \mathbb R^m . Generalizing the classical Martinelli–Bochner … Various types of solutions of the systems (D_x + iD_y)f = 0 are considered, where D_x and D_y are Dirac type operators in \mathbb R^m . Generalizing the classical Martinelli–Bochner formula for holomorphic functions, such a formula is proved for the C_1 -solutions of this system. Martinelli–Bochner formulae are also obtained for other overdetermined systems occuring in Clifford analysis.

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Plane wave decompositions of monogenic functions

1988-01-01

F. Sommen

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The inverse Fueter mapping theorem in integral form using spherical monogenics

2012-07-02

Fabrizio Colombo , Irene Sabadini , F. Sommen

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Fundamental solutions for operators which are polynomials in the Dirac operator

1992-01-01

F. Sommen , Xu Zhenyuan

A Clifford analysis approach to superspace

2007-04-15

Hendrik De Bie , F. Sommen

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Integration in superspace using distribution theory

2009-09-11

Kevin Coulembier , Hendrik De Bie , F. Sommen

In this paper, a new class of Cauchy integral formulae in superspace is obtained, using formal expansions of distributions. This allows to solve five open problems in the study of … In this paper, a new class of Cauchy integral formulae in superspace is obtained, using formal expansions of distributions. This allows to solve five open problems in the study of harmonic and Clifford analysis in superspace.

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Discrete Clifford Analysis: A Germ of Function Theory

2008-01-01

Fred Brackx , Hennie De Schepper , F. Sommen +1 more

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the … We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes' theorem, Cauchy's theorem and a Cauchy integral formula are established.

Hermite and Gegenbauer polynomials in superspace using Clifford analysis

2007-08-07

Hendrik De Bie , F. Sommen

The Clifford–Hermite and the Clifford–Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one … The Clifford–Hermite and the Clifford–Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an integration theory in superspace. Furthermore, a lot of basic properties, such as orthogonality relations, differential equations and recursion formulae, are proven. Finally, an interesting physical application of the super Clifford–Hermite polynomials is discussed, thus giving an interpretation to the super-dimension.

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Orthosymplectically invariant functions in superspace

2010-08-01

Kevin Coulembier , Hendrik De Bie , F. Sommen

The notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced. This leads to dimensional reduction theorems for differentiation and integration in superspace. These spherically symmetric … The notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced. This leads to dimensional reduction theorems for differentiation and integration in superspace. These spherically symmetric functions can be used to solve orthosymplectically invariant Schrödinger equations in superspace, such as the (an)harmonic oscillator or the Kepler problem. Finally, the obtained machinery is used to prove the Funk–Hecke theorem and Bochner’s relations in superspace.

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Introduction to Clifford analysis

2002-11-10

F. Sommen , Wolfgang Sprößig

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Fundamental solutions for the super Laplace and Dirac operators and all their natural powers

2007-06-21

Hendrik De Bie , F. Sommen

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Spherical monogenics on the Lie sphere

1990-09-01

F. Sommen

A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS

2011-01-01

Helmuth R. Malonek , Dixan Peña Peña , F. Sommen

In this paper we prove a Cauchy-Kowalevski theorem for the functions satisfying the system ∂xf∂x = 0 (called inframonogenic functions). In this paper we prove a Cauchy-Kowalevski theorem for the functions satisfying the system ∂xf∂x = 0 (called inframonogenic functions).

Clifford Analysis and Integral Geometry

1992-01-01

F. Sommen

Monogenic differential calculus

1991-01-01

F. Sommen

In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute … In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript m"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {R}^m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as residues of special differential forms. Next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de Rham cohomology spaces. We also study duals of spaces of monogenic differential forms, leading to a general residue theory in Euclidean space. Our theory includes the one established in our paper [11] and is strongly related to certain differential forms introduced by Habetha in [4].

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Twisted Plane Wave Expansions Using Hypercomplex Methods

2014-02-13

Fabrizio Colombo , Irene Sabadini , F. Sommen +1 more

The purpose of this paper is to derive various representations of the Dirac delta distribution, including a Bony-type twisted Radon decomposition, from boundary values of monogenic functions. This leads to … The purpose of this paper is to derive various representations of the Dirac delta distribution, including a Bony-type twisted Radon decomposition, from boundary values of monogenic functions. This leads to a new and simpler approach based on the properties of the analogue of the Cauchy kernel in the context of monogenic functions.

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An extension of the radon transform to clifford analysis

1987-06-01

F. Sommen

Using the decomposition of the elementary solution of the Dirac operator in plane wave type monogenic functions, we obtain an extension of the classical Radon transform in Euclidean space to … Using the decomposition of the elementary solution of the Dirac operator in plane wave type monogenic functions, we obtain an extension of the classical Radon transform in Euclidean space to the analytic functional of Clifford analysis. This includes, as a special case, the Radon transform of compactly supported hyperfunctions.

Fueter polynomials in discrete Clifford analysis

2011-09-08

Hilde De Ridder , Hennie De Schepper , F. Sommen

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SPHERICAL MEANS, DISTRIBUTIONS AND CONVOLUTION OPERATORS IN CLIFFORD ANALYSIS

2003-04-01

Fred Brackx , Richard Delanghe , F. Sommen

Axial Monogenic Functions from Holomorphic Functions

1993-11-01

A.K. Common , F. Sommen

Hypercomplex fourier and laplace transforms II

1983-02-01

F. Sommen

Abstract In this paper the exponential function introduced in [14] and the Cauchy kernel introduced in [7] are used to study new kinds of Fourier, Laplace, Cauchy and Hilbert transforms … Abstract In this paper the exponential function introduced in [14] and the Cauchy kernel introduced in [7] are used to study new kinds of Fourier, Laplace, Cauchy and Hilbert transforms of A valued L 2-functions in , where A is a Clifford algebra. In this way the corresponding theory in the complex plane (see [1], [2]) is generalized in the hypercomplex setting to several dimensions. AMS (MOS): 3063542B1044A1044A1546F20 ∗Research Assistant of the National Fund for Scientific Research (Belgium). ∗Research Assistant of the National Fund for Scientific Research (Belgium). Notes ∗Research Assistant of the National Fund for Scientific Research (Belgium).

Segal-Bargmann-Fock modules of monogenic functions

2017-10-01

Dixan Peña Peña , Irene Sabadini , F. Sommen

In this paper, we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extending it to Clifford algebra-valued functions. Then we apply the results to monogenic … In this paper, we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extending it to Clifford algebra-valued functions. Then we apply the results to monogenic functions and prove that the Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform for monogenic functionals. This kernel is also the reproducing kernel for the monogenic Bargmann module.

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On the singular Bochner-Martinelli integral

1998-09-01

R. Rocha-Ch�vez , Michael Shapiro , F. Sommen

Clifford analysis in two and several vector variables

1999-10-01

F. Sommen

In this paper we present some basic function theoretic results and ideas concerning nullsolutions of several Dirac type operators. In particular we consider plane wave solutions, axial symmetry, zonal functions … In this paper we present some basic function theoretic results and ideas concerning nullsolutions of several Dirac type operators. In particular we consider plane wave solutions, axial symmetry, zonal functions and irreducible representations of the group SO (m).

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Hermitean Clifford-Hermite Polynomials

2007-04-13

Fred Brackx , Hennie De Schepper , Nele De Schepper +1 more

SO(m)-invariant differential operators on Clifford algebra-valued functions

1993-11-01

F. Sommen , N. Van Acker

Reproducing bergman kernels in clifford analysis

1994-03-01

Fred Brackx , F. Sommen , N. Van Acker

Abstract It is proved that the module of Clifford-algebra-valued square-integrable eigenfunctions of the Dirac-operator in an open subset Ω of R m is a Hilbert-module with reproducing kernel; this reproducing … Abstract It is proved that the module of Clifford-algebra-valued square-integrable eigenfunctions of the Dirac-operator in an open subset Ω of R m is a Hilbert-module with reproducing kernel; this reproducing kernel for the case where Ω is the unit ball or the Euclidean space itself, is explicitly constructed. Also the module of square-integrable polymonogenic functions in R m is studied. It turns out that it is a Hilbert-module with reproducing kernel too. AMS No: 3003546E20 ∗Research associate supported by N.F.W.O., Belgium. ∗Research associate supported by N.F.W.O., Belgium. Notes ∗Research associate supported by N.F.W.O., Belgium.

q-deformed harmonic and Clifford analysis and theq-Hermite and Laguerre polynomials

2010-02-26

Kevin Coulembier , F. Sommen

We define a q-deformation of the Dirac operator, inspired by the one-dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we … We define a q-deformation of the Dirac operator, inspired by the one-dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows us to construct q-deformed Schrödinger equations in higher dimensions. The equivalence of these Schrödinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford–Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal with respect to an m-dimensional q-integration, which is related to integration on q-Euclidean space. The q-Laguerre polynomials are the eigenvectors of an suq(1|1)-representation.

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Szegö-Radon transform for hypermonogenic functions

2021-09-24

Alí Guzmán Adán , Ren Hu , T. Raeymaekers +1 more

Discrete weierstrass transform in discrete hermitian clifford analysis

2020-09-15

Alexandre Blondin Massé , F. Sommen , Hilde De Ridder +1 more

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Bargmann–Radon transform for axially monogenic functions

2019-11-03

Alí Guzmán Adán , Ren Hu , T. Raeymaekers +1 more

In this paper, we study the Bargmann–Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann–Radon transform for axially … In this paper, we study the Bargmann–Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann–Radon transform for axially monogenic functions, by making use of the Funk–Hecke theorem. Then we present the explicit form of the general Cauchy–Kowalewski extension for radial function. Finally, by making use of the results we obtained, we give an application of the Bargmann–Radon transform for Cauchy–Kowalewski extension.

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Szegö–Radon Transform for Biaxially Monogenic Functions

2019-09-06

Ren Hu , T. Raeymaekers , F. Sommen

Hypermonogenic solutions and plane waves of the Dirac operator in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>q</mml:mi></mml:msup></mml:mrow></mml:math>

2018-10-28

Alí Guzmán Adán , Heikki Orelma , F. Sommen

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Radon-Type Transforms for Holomorphic Functions in the Lie Ball

2018-09-24

Irene Sabadini , F. Sommen

Segal-Bargmann-Fock modules of monogenic functions

2017-10-01

Dixan Peña Peña , Irene Sabadini , F. Sommen

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Hypermonogenic solutions and plane waves of the Dirac operator in Rp x Rq

2017-09-18

Alí Guzmán Adán , Heikki Orelma , F. Sommen

In this paper we first define hypermonogenic solutions of the Dirac operator in Rp x Rq and study some basic properties, e.g., obtaining a Cauchy integral formula in the unit … In this paper we first define hypermonogenic solutions of the Dirac operator in Rp x Rq and study some basic properties, e.g., obtaining a Cauchy integral formula in the unit hemisphere. Hypermonogenic solutions form a natural function class in classical Clifford analysis. After that, we define the corresponding hypermonogenic plane wave solutions and deduce explicit methods to compute these functions.

Fueter’s Theorem for Monogenic Functions in Biaxial Symmetric Domains

2017-08-08

Dixan Peña Peña , Irene Sabadini , F. Sommen

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Plane wave formulas for spherical, complex and symplectic harmonics

2017-07-10

Hendrik De Bie , F. Sommen , Michael Wutzig

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On the Bargmann–Radon transform in the monogenic setting

2017-07-05

Fabrizio Colombo , Irene Sabadini , F. Sommen

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The Spingroup and its actions in discrete Clifford analysis

2017-01-29

Hilde De Ridder , F. Sommen

Recently, it has been established that the discrete star Laplace and the discrete Dirac operator, i.e. the discrete versions of their continuous counterparts when working on the standard grid, are … Recently, it has been established that the discrete star Laplace and the discrete Dirac operator, i.e. the discrete versions of their continuous counterparts when working on the standard grid, are rotation-invariant. This was done starting from the Lie algebra so(m, C) corresponding to the special orthogonal Lie group SO(m), considering its representation in the discrete Clifford algebra setting and proving that these operators are symmetries of the Dirac and Laplace operators. This set-up showed in an abstract way that representation-theoretically the discrete setting mirrors the Euclidean Clifford analysis setting. However from a practical point of view, the group-action remains indispensable for actual calculations. In this paper, we define the discrete Spingroup, which is a double cover of SO(m), and consider its actions on discrete functions. We show that this group-action makes the spaces Hk and Mk into Spin(m)-representations. We will often consider the compliance of our results to the results under the so(m, C)- action.

The Spingroup and its actions in discrete Clifford analysis

2017-01-01

Hilde De Ridder , F. Sommen

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Hypermonogenic solutions and plane waves of the Dirac operator in Rp x Rq

2017-01-01

Alí Guzmán Adán , Heikki Orelma , F. Sommen

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Fueter's theorem for monogenic functions in biaxial symmetric domains

2016-11-04

Dixan Peña Peña , Irene Sabadini , F. Sommen

In this paper we generalize the result on Fueter's theorem from [10] by Eelbode et al. to the case of monogenic functions in biaxially symmetric domains. To obtain this result, … In this paper we generalize the result on Fueter's theorem from [10] by Eelbode et al. to the case of monogenic functions in biaxially symmetric domains. To obtain this result, Eelbode et al. used representation theory methods but their result also follows from a direct calculus we established in our paper [21]. In this paper we first generalize [21] to the biaxial case and derive the main result from that.

A Fourier–Borel transform for monogenic functionals

2016-10-01

Irene Sabadini , F. Sommen

We discuss the Fourier–Borel transform for the dual of spaces of monogenic functions. This transform may be seen as a restriction of the classical Fourier–Borel transform for holomorphic functionals, and … We discuss the Fourier–Borel transform for the dual of spaces of monogenic functions. This transform may be seen as a restriction of the classical Fourier–Borel transform for holomorphic functionals, and it transforms spaces of monogenic functionals into quotients of spaces of entire holomorphic functions of exponential type. We prove that, for the Lie ball, these quotient spaces are isomorphic to spaces of monogenic functions of exponential type.

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Rotations in discrete Clifford analysis

2016-04-13

Hilde De Ridder , T. Raeymaekers , F. Sommen

Reproducing Kernels for Polynomial Null-Solutions of Dirac Operators

2016-02-16

Hendrik De Bie , F. Sommen , Michael Wutzig

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Differential Forms and Clifford Analysis

2016-01-01

Irene Sabadini , F. Sommen

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Modern Trends in Hypercomplex Analysis

2016-01-01

Swanhild Bernstein , Uwe Kähler , Irene Sabadini +1 more

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Examples of Morphological Calculus

2016-01-01

F. Sommen

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Script Geometry

2016-01-01

Paula Cerejeiras , Uwe Kähler , F. Sommen +1 more

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Fueter's theorem for monogenic functions in biaxial symmetric domains

2016-01-01

Dixan Peña Peña , Irene Sabadini , F. Sommen

On the Szegő–Radon projection of monogenic functions

2015-10-23

Fabrizio Colombo , Irene Sabadini , F. Sommen

Sato’s Hyperfunctions and Boundary Values of Monogenic Functions

2014-09-09

Irene Sabadini , F. Sommen , Daniele C. Struppa

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Twisted Plane Wave Expansions Using Hypercomplex Methods

2014-02-13

Fabrizio Colombo , Irene Sabadini , F. Sommen +1 more

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On solutions of a discretized heat equation in discrete Clifford analysis

2013-09-30

Franka Baaske , Swanhild Bernstein , Hilde De Ridder +1 more

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space … The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials.

Distributions and the Global Operator of Slice Monogenic Functions

2013-07-29

Fabrizio Colombo , F. Sommen

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Cauchy-Kowalevski extensions and monogenic plane waves using spherical monogenics

2013-06-01

Nele De Schepper , F. Sommen

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Clifford superanalysis

2013-01-01

F. Sommen

A survey of superanalysis with emphasis on superforms, superchains, superboundaries and integration is given. Moreover, the basic concepts for Clifford analysis on superspace, including the super-Dirac operator and Cauchy's integral … A survey of superanalysis with emphasis on superforms, superchains, superboundaries and integration is given. Moreover, the basic concepts for Clifford analysis on superspace, including the super-Dirac operator and Cauchy's integral formula, are given and the calculus of Clifford superforms, leading to a general Cauchy integral formula, is presented.

The inverse Fueter mapping theorem in integral form using spherical monogenics

2012-07-02

Fabrizio Colombo , Irene Sabadini , F. Sommen

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Cauchy-type theorems on surfaces

2012-01-01

F. Sommen

A Cauchy–type theorem for the tangential Dirac operator on a surface embedded in Euclidean space is proved by two methods: a distributional one and by using differential forms. Moreover it … A Cauchy–type theorem for the tangential Dirac operator on a surface embedded in Euclidean space is proved by two methods: a distributional one and by using differential forms. Moreover it is shown how both methods are linked to each other.

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Diffusive wavelets on the Spin group

2012-01-01

Swanhild Bernstein , Svend Ebert , F. Sommen

The first part of this article is devoted to a brief review of the results about representation theory of the spin group Spin(m) from the point of view of Clifford … The first part of this article is devoted to a brief review of the results about representation theory of the spin group Spin(m) from the point of view of Clifford analysis. In the second part we are interested in Clifford-valued functions and wavelets on the sphere. The connection of representations of Spin(m) and the concept of diffusive wavelets leads naturally to investigations of a modified diffusion equation on the sphere, that makes use of the Gamma operator. We will achieve to obtain Clifford-valued diffusion wavelets with respect to a modified diffusion operator. Since we are able to characterize all representations of Spin(m) and even to obtain all eigenvectors of the (by representation) regarded Casimir operator in representation spaces, it seems appropriate to look at functions on Spin(m) directly. Concerning this, our aim shall be to formulate eigenfunctions for the Laplace-Beltrami operator on Spin(m) and give the series expansion of the heat kernel on Spin(m) in terms of eigenfunctions.

Discrete Clifford analysis: the one-dimensional setting

2011-12-05

Hendrik De Bie , Hilde De Ridder , F. Sommen

In a higher dimensional setting, there are two major theories generalizing the theory of holomorphic functions in the complex plane, namely the theory of several complex variables and Clifford analysis. … In a higher dimensional setting, there are two major theories generalizing the theory of holomorphic functions in the complex plane, namely the theory of several complex variables and Clifford analysis. Discrete Clifford analysis is a discrete counterpart of the latter, studying the null functions of a discrete Dirac operator, which are called discrete monogenic functions. In this contribution, we give several new results in the one-dimensional case. We focus on the basic building blocks of discrete functions, namely discrete delta functions δ j , in relation to the discrete vector variable operator ξ. We introduce discrete distribution theory, in particular discrete delta distributions δ j and define a Fourier transform for discrete distributions. Finally, a comparison is made between discrete delta functions and distributions.

Fueter polynomials in discrete Clifford analysis

2011-09-08

Hilde De Ridder , Hennie De Schepper , F. Sommen

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Closed form of the Fourier–Borel Kernel in the Framework of Clifford Analysis

2011-05-10

Nele De Schepper , F. Sommen

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Operator Identities in q-Deformed Clifford Analysis

2011-03-02

Kevin Coulembier , F. Sommen

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Micro-localisation from Clifford Analysis

2011-01-01

F. Sommen , Theodore E. Simos , George Psihoyios +2 more

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite … Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation F. Sommen; Micro‐localisation from Clifford Analysis. AIP Conf. Proc. 22 September 2011; 1389 (1): 17–19. https://doi.org/10.1063/1.3636660 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS

2011-01-01

Helmuth R. Malonek , Dixan Peña Peña , F. Sommen

The class of Clifford-Fourier transforms

2011-01-01

Hendrik De Bie , Nele De Schepper , F. Sommen

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 … Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see [F. Brackx et al., The Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.

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A Hilbert Transform for Matrix Functions on Fractal Domains

2010-11-18

Ricardo Abreu Blaya , Juan Bory Reyes , Fred Brackx +2 more

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Harmonic and Monogenic Functions in Superspace

2010-11-17

Kevin Coulembier , Hendrik De Bie , F. Sommen

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Orthosymplectically invariant functions in superspace

2010-08-01

Kevin Coulembier , Hendrik De Bie , F. Sommen

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A Hermitian Cauchy formula on a domain with fractal boundary

2010-03-25

Ricardo Abreu Blaya , Juan Bory Reyes , Fred Brackx +2 more

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q-deformed harmonic and Clifford analysis and theq-Hermite and Laguerre polynomials

2010-02-26

Kevin Coulembier , F. Sommen

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The Cauchy-Kovalevskaya Extension Theorem in Discrete Clifford Analysis

2010-01-01

Hilde De Ridder , Hennie De Schepper , F. Sommen +3 more

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic … Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy‐Kovalevskaya extension theorem for discrete monogenic functions defined on the grid Zhm of m‐tuples of integer multiples of a variable mesh width h. Convergence to the continuous case is investigated. As illustrative examples we explicitly construct the Cauchy‐Kovalevskaya extensions of the discrete delta function and of a discretized exponential.

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Integration in superspace using distribution theory

2009-09-11

Kevin Coulembier , Hendrik De Bie , F. Sommen

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On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

2009-09-01

Fred Brackx , Bram De Knock , Hennie De Schepper +1 more

Spherical harmonics and integration in superspace: II

2009-05-28

Hendrik De Bie , David Eelbode , F. Sommen

The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007 The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007

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A Cauchy integral formula in superspace

2009-05-22

Hendrik De Bie , F. Sommen

In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework … In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal Clifford analysis. After introducing Clifford algebra-valued surface- and volume-elements, a purely fermionic Cauchy formula is proved. Combining this formula with the already well-known bosonic Cauchy formula yields the general case. Here the integration over the boundary of a supermanifold is an integration over the even as well as the odd boundary (in a formal way). Finally, some additional results such as a Cauchy–Pompeiu formula and a representation formula for monogenic functions are proved.

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Coauthor	Papers Together
Irene Sabadini	21
Fred Brackx	20
Richard Delanghe	14
Hendrik De Bie	14
Nele De Schepper	12
Fabrizio Colombo	12
Hennie De Schepper	11
David Eelbode	10
Vladimı́r Souček	10
Daniele C. Struppa	9
Hilde De Ridder	8
N. Van Acker	7
Dixan Peña Peña	7
Kevin Coulembier	5
Uwe Kähler	5
T. Raeymaekers	5
Paula Cerejeiras	5
Alí Guzmán Adán	5
P. Van Lancker	4
George Psihoyios	4
Theodore E. Simos	4
Ch. Tsitouras	4
Ren Hu	3
A.K. Common	3
Heikki Orelma	3
Swanhild Bernstein	3
Jarolím Bureš	2
Ricardo Abreu Blaya	2
Wolfgang Sprößig	2
Michael Shapiro	2
Michael Wutzig	2
Tao Qian	2
Kit Ian Kou	2
Juan Bory Reyes	2
Bram De Knock	2
Svend Ebert	1
Denis Constales	1
Gil Bernardes	1
Zacharias Anastassi	1
Xu Zhenyuan	1
R. Rocha-Ch�vez	1
Gerhard Jank	1
C. J. L. Doran	1
Souček	1
David Hestenes	1
Helmuth R. Malonek	1
N. Vieira	1
Franka Baaske	1
Reynaldo Rocha-Chávez	1
Adrian Vajiac	1

Clifford Algebra and Spinor-Valued Functions

1992-01-01

Richard Delanghe , F. Sommen , Vladimı́r Souček

Clifford algebra and spinor-valued functions (a function theory for the Dirac operator)

1992-11-01

Richard Delanghe , F. Sommen , Vladimı́r Souček

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Clifford Algebra to Geometric Calculus

1984-01-01

David Hestenes , Garret Sobczyk

Special functions in Clifford analysis and axial symmetry

1988-02-01

F. Sommen

Analysis of Dirac Systems and Computational Algebra

2004-01-01

Fabrizio Colombo , Irene Sabadini , F. Sommen +1 more

An extension of the radon transform to clifford analysis

1987-06-01

F. Sommen

Quaternionic and Clifford calculus for physicists and engineers

1997-11-01

Klaus Gürlebeck , Wolfgang Sprößig

Spherical harmonics and integration in superspace: II

2009-05-28

Hendrik De Bie , David Eelbode , F. Sommen

The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007 The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007

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Generalization of the Cauchy-Riemann Equations and Representations of the Rotation Group

1968-01-01

E. M. Stein , George H. Weiss

Correct Rules for Clifford Calculus on Superspace

2007-05-26

Hendrik De Bie , Frank Sommen

Clifford Analysis and Integral Geometry

1992-01-01

F. Sommen

Spherical Monogenic Functions and Analytic Functionals on the Unit Sphere

1981-12-01

Fransiscus SOMMEN

The aim of this paper is to introduce the analogues of the restrictions of $z\to z^k$, $k\in\mathbf{Z}$, $z\in\mathbf{C}$, to the unit circle, in the case of the unit sphere $\partial … The aim of this paper is to introduce the analogues of the restrictions of $z\to z^k$, $k\in\mathbf{Z}$, $z\in\mathbf{C}$, to the unit circle, in the case of the unit sphere $\partial B(0,1)$ in $\mathbf{R}^{m+1}$ and to apply them to boundary value problems of left monogenic functions in $\mathbf{R}^{m+1}\backslash \partial B(0,1)$.

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Spherical harmonics and integration in superspace

2007-06-12

Hendrik De Bie , F. Sommen

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Duality in hypercomplex function theory

1980-06-01

Richard Delanghe , Fred Brackx

A Clifford analysis approach to superspace

2007-04-15

Hendrik De Bie , F. Sommen

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Spherical monogenics on the Lie sphere

1990-09-01

F. Sommen

Discrete function theory based on skew Weyl relations

2010-09-01

Hilde De Ridder , Hennie De Schepper , Uwe Kähler +1 more

In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means of a new "skew" type of Weyl relations. We will show that … In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means of a new "skew" type of Weyl relations. We will show that this new type overcomes the difficulties of working with standard Weyl relations in the discrete case. A Fischer decomposition, Euler operator, monogenic projection, and basic homogeneous powers will be constructed.

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Clifford analysis, biaxial symmetry and pseudoanalytic functions

1990-02-01

Gerhard Jank , F. Sommen

In this paper we develop a plane system of first-order differential equations, describing nullsolutions of the Dirac operator in biaxially symmetric domains. The solutions to the equations are of pseudoanalytic … In this paper we develop a plane system of first-order differential equations, describing nullsolutions of the Dirac operator in biaxially symmetric domains. The solutions to the equations are of pseudoanalytic type and may be used to construct on orthogonal basis for the spaces of so-called spherical monogenics. which are related to spherical harmonics with spin. Finally, after certain restrictions on dimensins. We give a complete characterization of these pscudoanalytic functions in terms of holomorphic ones. This paper may thus ge considered as a bridge between Clifford analysis and the theory of pscudoanalytic functions.

Monogenic Differential Operators

1992-11-01

F. Sommen , N. Van Acker

On finite difference potentials and their applications in a discrete function theory

2002-11-10

Klaus Gürlebeck , Angela Hommel

Abstract We present a potential theoretical method which is based on the approximation of the boundary value problem by a finite difference problem on a uniform lattice. At first the … Abstract We present a potential theoretical method which is based on the approximation of the boundary value problem by a finite difference problem on a uniform lattice. At first the discrete fundamental solution of the Laplace equation is studied and the theory of difference potentials is described. In the second part we define a discrete Cauchy integral operator and a Teodorescu transform. In addition a Borel–Pompeiu formula can be formulated. Copyright © 2002 John Wiley & Sons, Ltd.

Fundaments of Hermitean Clifford Analysis Part I: Complex Structure

2007-03-14

Fred Brackx , Jarolím Bureš , Hennie De Schepper +3 more

AN ALGEBRA OF ABSTRACT VECTOR VARIABLES

1997-01-01

F. Sommen

A product and an exponential function in hypercomplex function theory

1981-01-01

F. Sommen

It is proved that any A-valued analytic function in an open subset Ω of Rm may be extended to a monogenic function in a suitable open neighbourhood Ωof Ω in … It is proved that any A-valued analytic function in an open subset Ω of Rm may be extended to a monogenic function in a suitable open neighbourhood Ωof Ω in Rm+1. This result is applied to define a product between monogenic functions, which in its turn is used to determine a hypercomplex exponential function

Groups and Geometric Analysis

2000-10-03

Sigurđur Helgason

Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces … Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces Appendix Some details Bibliography Symbols frequently used Index Errata.

Monogenic Differential Calculus

1991-08-01

F. Sommen

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Taylor Series Expansion in Discrete Clifford Analysis

2013-04-11

Hilde De Ridder , Hennie De Schepper , Frank Sommen

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Quaternionic analysis

1979-03-01

A. Sudbery

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely … The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself (1) and his principal followers and expositors, Tait(2) and Joly(3), only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.

Explicit solutions of the inhomogeneous dirac equation

1997-12-01

Frank Sommen , Bernard Jancewicz

On the Szegő–Radon projection of monogenic functions

2015-10-23

Fabrizio Colombo , Irene Sabadini , F. Sommen

Hermitian Clifford analysis and resolutions

2002-11-10

Irene Sabadini , Frank Sommen

Abstract In this paper, we discuss the so‐called Witt basis in a Clifford algebra and we axiomatically define an algebra of abstract Hermitian vector variables similar to the ‘radial algebra’. … Abstract In this paper, we discuss the so‐called Witt basis in a Clifford algebra and we axiomatically define an algebra of abstract Hermitian vector variables similar to the ‘radial algebra’. In this setting, we introduce some linear partial differential operators and we study their resolutions. Copyright © 2002 John Wiley & Sons, Ltd.

On a Generalization of Fueter’s Theorem

2000-12-31

F. Sommen

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Fundaments of Hermitean Clifford analysis part II: splitting ofh-monogenic equations

2007-10-01

Fred Brackx , Jarolím Bureš , Hennie De Schepper +3 more

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A global theory of supermanifolds

1980-06-01

Alice Rogers

A mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds … A mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds to superfields. This new construction is compared with several pre-existing definitions of supermanifold and graded manifold; it is shown to include all these definitions and to go beyond them, particularly in admitting the possibility of nontrivial topology in the anticommuting sector. Local differential geometry and potential applications to supergravity are considered.

Discrete Dirac Operators in Clifford Analysis

2007-05-23

Nelson Faustino , Uwe Kähler , Frank Sommen

A Basic Framework for Discrete Clifford Analysis

2009-01-01

Hennie De Schepper , Frank Sommen , Liesbet Van de Voorde

Abstract A basic framework is derived for the development of a higher-dimensional discrete function theory in a Clifford algebra context. The concept of a discrete monogenic function is introduced as … Abstract A basic framework is derived for the development of a higher-dimensional discrete function theory in a Clifford algebra context. The concept of a discrete monogenic function is introduced as a proper generalization of the discrete holomorphic, or monodiffric, functions introduced by Isaacs in the 1950s. A concrete model is provided for the definition of the corresponding discrete Dirac operator. Keywords: Primary 30G35Clifford analysisdiscrete Dirac operatordiscrete monogenic function

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Applications of Grassmann's Extensive Algebra

1878-01-01

Professor Clifford

Quaternionic Analysis and Elliptic Boundary Value Problems

1990-01-01

Klaus Gürlebeck , Wolfgang Sprößig

1. Quaternionic Analysis.- 1.1. Algebra of Real Quaternions.- 1.2. H-regular Functions.- 1.3. A Generalized LEIBNIZ Rule.- 1.4. BOREL-POMPEIU's Formula.- 1.5. Basic Statements of H-regular Functions.- 2. Operators.- 2.3. Properties of … 1. Quaternionic Analysis.- 1.1. Algebra of Real Quaternions.- 1.2. H-regular Functions.- 1.3. A Generalized LEIBNIZ Rule.- 1.4. BOREL-POMPEIU's Formula.- 1.5. Basic Statements of H-regular Functions.- 2. Operators.- 2.3. Properties of the T-Operator.- 2.4. VEKUA's Theorems.- 2.5. Some Integral Operators on the Manifold.- 3. Orthogonal Decomposition of the Space L2,H(G).- 4. Some Boundary Value Problems of DIRICHLET's Type.- 4.1. LAPLACE Equation.- 4.2. HELMHOLTZ Equation.- 4.3. Equations of Linear Elasticity.- 4.4. Time-independent MAXWELL Equations.- 4.5. STOKES Equations.- 4.6. NAVIER-STOKES Equations.- 4.7. Stream Problems with Free Convection.- 4.8. Approximation of STOKES Equations by Boundary Value Problems of Linear Elasticity.- 5. H-regular Boundary Collocation Methods.- 5.1. Complete Systems of H-regular Functions.- 5.2. Numerical Properties of H-complete Systems of H-regular Functions.- 5.3. Foundation of a Collocation Method with H-regular Functions for Several Elliptic Boundary Value Problems.- 5.4. Numerical Examples.- 6. Discrete Quaternionic Function Theory.- 6.1. Fundamental Solutions of the Discrete Laplacian.- 6.2. Fundamental Solutions of a Discrete Generalized CAUCHY-RIEMANN Operator.- 6.3. Elements of a Discrete Quaternionic Function Theory.- 6.4. Main Properties of Discrete Operators.- 6.5. Numerical Solution of Boundary Value Problems of NAVIER-STOKES Equations.- 6.6. Concluding Remarks.- References.- Notations.

Analytic Functionals on the Lie Sphere

1980-06-01

Mitsuo Morimoto

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Spingroups and Spherical Monogenics

1986-01-01

Richard Delanghe , F. Sommen

Clifford Analysis: History and Perspective

2001-09-01

Richard Delanghe

Radon and X-ray transforms in clifford analysis

1989-02-01

F. Sommen

Starting from the decompositions of the elementary solution of a Dirac-type operator in in terms of similar solutions in lower dimensions, we obtain boundary value representations for the Radon transforms … Starting from the decompositions of the elementary solution of a Dirac-type operator in in terms of similar solutions in lower dimensions, we obtain boundary value representations for the Radon transforms of functions in of arbitrary codimension. As a special case, we obtain representations for both the classical Radon transform and the X-ray transform. Next, our theory is extended to the analytic functional of Clifford analysis, thus including compactly supported hyperfunctions. We also prove a Paley-Wiener-Schwartz type theorem for the radial pan of these Clifford - Radon transforms. Finally we construct a transform, mapping analytic functionals into holomorphic functions, which may be expressed in terms of integrals of functions over concentric cylinders.

Axial Monogenic Functions from Holomorphic Functions

1993-11-01

A.K. Common , F. Sommen

The Hermitian Clifford Analysis Toolbox

2008-04-26

Fred Brackx , Hennie De Schepper , Frank Sommen

The generalized clifford-hermite continuous wavelet transform

2001-02-01

Fred Brackx , F. Sommen

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Complexified clifford analysis

1982-09-01

John Ryan

The foundations of a function theory, in several complex variables, over complex Clifford algebras are developed The influence within this theory of complex analysis, in one variable, is demonstrated. A … The foundations of a function theory, in several complex variables, over complex Clifford algebras are developed The influence within this theory of complex analysis, in one variable, is demonstrated. A generalized Cauchy integral formula is provided, and complex harmonic functions are used to construct holomorphic functions which satisfy a generalized Cauchy Riemann equation introduced here.

Some connections between clifford analysis and complex analysis

1982-09-01

F. Sommen

INTRODUCTION TO THE THEORY OF SUPERMANIFOLDS

1980-02-28

Dimitry Leites

CONTENTS Introduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § … CONTENTS Introduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § 6. The supertrace § 7. The Berezinian (Berezin function) § 8. Tensor algebras § 9. Lie superalgebras and derivations of superalgebras Chapter II. Analysis in superspaces and superdomains § 1. Definition of superspaces and superdomains § 2. Vector fields and Taylor series § 3. The inverse function theorem and the implicit function theorem § 4. Integration in superdomains Chapter III. Supermanifolds § 1. Definition of a supermanifold § 2. Subsupermanifolds § 3. Families Notes References

Spingroups and Spherical Means

1986-01-01

F. Sommen

Special Functions and the Theory of Group Representations

1968-12-31

N. I︠a︡. Vilenkin

A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of … A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special functions are related to (and derived from) simple well-known facts of representation theory. The book combines the majority of known results in this direction. In particular, the author describes connections between the exponential functions and the additive group of real numbers (Fourier analysis), Legendre and Jacobi polynomials and representations of the group $SU(2)$, and the hypergeometric function and representations of the group $SL(2,R)$, as well as many other classes of special functions.

Monogenic differential calculus

1991-01-01

F. Sommen

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