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The main goal of this paper is to give the basic principles of hyperbolic function theory. First, we discuss the Green identity for H k ¯ H k . Second, we also deal with the integral representations for hyperbolic harmonic function in the Clifford algebra C l n + 1 , 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider general multivector elements of Clifford algebras C l (3 , 0) , C l (1 , 2) and C l (0 , 3) , and look for possibilities to factorize multivectors into products of blades, idempotents and exponentials, where the exponents are frequently blades of grades zero (scalar) to n (pseudoscalar). [ABSTRACT FROM AUTHOR]

In the classical complex analysis of several variables Lars Hoermander had been proven for Hartog's extension theorem. His proof is based on solving the inhomogeneous Cauchy-Riemann system. However, it is difficult when generalizing this diagram to prove this problem in Clifford analysis because of the non-commutative property. In this paper we will propose a method to overcome this difficultly and we will prove the Hartog's extension theorem for multi-regular function taking valued in Clifford algebra. [ABSTRACT FROM AUTHOR]

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here, we consider polynomials of the form P (x) = ∑ k = 0 d x k a k , with Clifford coefficients a k ∈ R n , and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy–Riemann) operator and extend the results to slice-regular functions. [ABSTRACT FROM AUTHOR]

John Ryan is seen as a leading expert in Clifford analysis. The short communication gives an insight in his career, his scientific fields and his efforts. [ABSTRACT FROM AUTHOR]

We introduce and study Hardy spaces consisting of Clifford algebra-valued functions annihilated by perturbed Dirac operators in exterior uniformly rectifiable (UR) domains, and which radiate at infinity. In this context, we establish a higher dimensional version of Cauchy’s vanishing theorem, whose proof makes use of the properties of Cauchy-like operators in exterior UR domains, a sharp version of the Divergence Theorem in exterior Ahlfors regular domains, and a good understanding of the nature of various radiation conditions and properties of the far field pattern for Clifford algebra-valued null-solutions of the Helmholtz operator. [ABSTRACT FROM AUTHOR]

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on representation theory of Lie groups is the natural way to construct such a Dirac-type operator because applied to a Riemannian spin manifold, it provides the Atiyah–Singer Dirac operator. This method, however, does not apply to these Hermitian Dirac operators because the representations of the unitary group used are not irreducible, causing problems in considering invariance under a group larger thanU(n). This motivates either the development of Clifford analysis over a complex vector space with respect to a Hermitian inner product or the development of Dirac-type operators on Cauchy–Riemann structures. [ABSTRACT FROM AUTHOR]

We consider the notion of the Laurent series for the theory of bicomplex holomorphic functions. Some basic properties of it are established. A special attention is paid to a detailed description of the sets of convergence and divergence of such series which reveals many peculiarities of the situation. [ABSTRACT FROM AUTHOR]

We describe the relationship between the growth conditions of monogenic extensions of square-integrable functionsfin terms of pointwise bounds or bounds on integral averages on the one hand, and the support of the Fourier transformor its annihilation by certain higher-dimensional analogues of the signum function on the other. We review known results involving a function’s monogenic extension and their classical Fourier transform. These results are extended to the Clifford-Fourier transform of Brackx, De Schepper and Sommen. The equivalence of the pointwise bounds and the bounds on the integral averages is observed as a consequence. [ABSTRACT FROM AUTHOR]

Unlike in complex analysis, in all hypercomplex function theories a hypercomplex variable is not monogenic (regular) and there exists a problem to define analogues of positive and negative powers of the complex variable. R. Fueter firstly introduces a system of symmetric regular quaternion polynomials as analogues of positive powers of a complex variable and proves the Taylor theorem in his theory. In Clifford analysis an analogical idea is used. The Fueter symmetric polynomials are both left- and right-regular, the symmetric polynomials in Clifford analysis are also both left- and right-monogenic. In this paper we construct only left-regular quaternion polynomials and show that the theory of regular quaternion functions of a vector valued quaternion variable can be developed using these polynomials. [ABSTRACT FROM AUTHOR]

We consider the modified Cauchy–Riemann operatorin the universal Clifford algebrawith the basis. The null-solutions of this operator are calledk-hypermonogenic functions. We calculate thek-hyperbolic harmonic fundamental solutions, i.e. solutions to, and use these solutions to findk-hypermonogenic kernels for a Cauchy-type integral formula in the upper half-space. [ABSTRACT FROM AUTHOR]

In this paper, we give the definition of T-operator with bihypermonogenic kernel in Clifford analysis and discuss a series of properties of this operator, such as uniform boundness, Hölder continuity and- integrability. T-operator is a singular integral operator which is defined in then-dimensional Euclidean space valued in the noncommutative Clifford algebra. The properties of T-operator play an important role in solving differential equations. [ABSTRACT FROM AUTHOR]

First, three important theorems related to Clifford Möbius transformations in the real Cliffordare given. Then it is proved that the composition of ak-hypergenic function with a Clifford Möbius transformation leads to ak-hypergenic function with weight, where. Finally, the relationships betweenk-hypergenic functions and-hypergenic functions are obtained, where. [ABSTRACT FROM PUBLISHER]

We study the generalized Beltrami equation, whereis the left Dirac operator inacting on functions inand with values in the complex Clifford algebra,is its conjugate, andis a-valued function with compact support, with vanishing mean oscillation, satisfying, whereare the coordinates ofin. Letbe a weight function in. We prove that ifbelongs to the Muckenhoupt classwith,that makes continuous the Riesz potential, then for every, there exists a solution of the equation above satisfyingfor every distributional partial derivative. If, we prove the same result for any. [ABSTRACT FROM PUBLISHER]

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries. [ABSTRACT FROM AUTHOR]

The theory of split-quaternion analysis and split-quaternion geometry is nowadays under full development. In this article, we extend quaternionic Hermitian Clifford analysis to the case of split-quaternions. The split-quaternionic Hermitian Dirac operatoris introduced as a self-map of smooth functions defined in domains ofwith values in the tensor product of the hyperbolic quaternions, the split-quaternionsand the Clifford algebra. It is defined bywithbeing the basis ofanddenoting the twisted Hermitian Dirac operators in the split-quaternionic Clifford algebrawhose definition is based on a delicate construction of the split-quaternionic Witt basis. We establish the counterpart of the Cauchy integral formula. [ABSTRACT FROM AUTHOR]

In this article, we obtain an explicit expression of the difference and sum of the unit base in the Clifford analysis case by using the analogy with the complex one. These formulae then lead to the introduction of a class of Bernoulli polynomials in the Clifford analysis setting which are linked to the sum base. The newly introduced Bernoulli polynomials have properties which are very similar to those of the classical ones. Then we prove that each of the sum and difference of the unit base is of order 1. Finally, certain convergence properties of the difference and sum of simple monic bases of special monogenic polynomials are investigated. The obtained results generalize the complex case proved by Mikhail and Nassif to the Clifford setting. [ABSTRACT FROM PUBLISHER]

The aim of this article is to give some boundary properties of hypergenic quasi-Cauchy type integrals in Clifford analysis. First, the Plemelj formula for hypergenic quasi-Cauchy type integrals is given by virtue of the Cauchy integral formula for hypergenic functions. Then, the Privalov theorem for hypergenic quasi-Cauchy type integrals is obtained by virtue of the Plemelj formula. [ABSTRACT FROM PUBLISHER]

In this article, some properties of singular integrals with parameter are discussed. We prove a Liouville-type theorem for pseudo-monogenic functions and pseudo-harmonic functions. With the help of them,R−1Riemann boundary value problems for pseudo-monogenic functions and pseudo-harmonic functions with values in a Clifford algebraCl(V3.3) are investigated in the classical sense and their explicit representation of solutions are obtained. [ABSTRACT FROM AUTHOR]

The equationDu = F(x, u) in Clifford algebras depending on parameters is considered, where D is the Cauchy–Riemann operator. A sufficient condition for the right-hand side of the equation to be anti-monogenic is given. The given criterion enables the construction of anti-monogenic functions in Clifford algebras depending on parameters and yields interiorLp-estimates as well. The results provide generalizations of those in Tutschke and Yüksel (1999) [W. Tutschke and U. Yüksel,Generalized Monogenic Functions Satisfying Differential Equations with Anti-Monogenic Right-Hand Sides, ISAAC Series, Vol. 6, Kluwer Academic Publishers, Dordrecht, 1999, pp. 263–270]. [ABSTRACT FROM AUTHOR]

We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For each suitable choice of a real vector subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case, we get a volume Cauchy formula. In the Clifford algebra case, the choice of the paravector subspace ℝn+1gives a volume Cauchy formula for slice monogenic functions. [ABSTRACT FROM AUTHOR]

First, we consider the relationships between complexk-hypermonogenic functions and complex monogenic functions. Then we discuss necessary and sufficient conditions for complexk-hypermonogenic functions which is similar to the Cauchy–Riemann equation. Finally, we establish three theorems about complex partial differential equations. [ABSTRACT FROM AUTHOR]

In this article, we first suggest an alternative approach to extend the original Fueter's theorem in Dunkl–Clifford analysis to a version of the higher order case. We then use this result to prove a generalized version of Fueter's theorem with an extra homogeneous Dunkl-monogenic polynomialPn(x0, x) ininstead of the classical factorPn(x) in ℝd. [ABSTRACT FROM AUTHOR]

Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, calledq-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article. [ABSTRACT FROM AUTHOR]

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed in the framework of circulant (2 × 2) matrix functions. Under this setting, we will present the half Dirichlet problem with boundary spaces of Hölder continuous circulant (2 × 2) matrix functions on the sphere of even-dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solutions to the Dirichlet problem on the unit ball for Hölder continuous circulant (2 × 2) matrix functions as the boundaries and the solutions to the classical Dirichlet problem as the special case, and derive a decomposition of the Poisson kernel for matrix Laplace operator. [ABSTRACT FROM AUTHOR]

We research a function theory in higher dimensions based on the hyperbolic metric. The complex numbers are extended by the Clifford algebraCl0,ngenerated by the anti-commutating elementseisatisfying. In 1992, H. Leutwiler noticed that the power function (x0 + x1e1 + ··· + xnen)mis the generalized conjugate gradient of the functions. In the complex field (n = 1) this functionhis harmonic in the usual sense, but in the higher dimensional case it is harmonic with respect to the Laplace–Beltrami operator with respect to the Riemannian hyperbolic metric. He started to study these type of functions, calledH-solutions, that include positive and negative powers and elementary functions. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined by H. Leutwiler and the first author in 2000. The integral formula has been proved by the first author. In this article, we present a simple way to find hyperbolic harmonic functions depending on the hyperbolic distance. We use these functions to determine a better presentation of the kernel, that is surprisingly the shifted Euclidean Cauchy kernel. We prove a power series expansion of hypermonogenic functions and present a version of the Maximum Modulus theorem. [ABSTRACT FROM AUTHOR]

In this article, we first give the Plemelj formula for functions with parameter by following the classical method. Then by using the higher order Cauchy integral representation formulas, some properties for harmonic functions and bi-harmonic functions are presented, for example, the mean value theorem, the Painlevé theorem, etc. Finally, we consider the Riemann boundary value problems for harmonic functions and bi-harmonic functions in Clifford analysis, the solutions are given in an explicit way. [ABSTRACT FROM AUTHOR]

We consider an initial value problem of typewheretis the time,x ∈ ℝn+1anduis a Clifford-type algebra-valued function, satisfying int = 0 the equation, λj(x) ∈ ℝ for allj. We will solve this problem using the theory of associated spaces. In order to do that, we give sufficient conditions on the coefficients of the operators ℱ andD, whenforA(i)(x) ∈ ℝ orA(i)(x) belonging to a Clifford-type algebra, such that these operators form an associated pair. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

In this article a Dirichlet problem in R3 is solved for functions which are monogenic with respect to a Clifford algebra depending on parameters. If the parameters of the Clifford algebra depend on the variable x ∈ R3, then the coefficients of the Cauchy-Riemann system for monogenic functions depend on x. [ABSTRACT FROM AUTHOR]

The main goal of this article is to study special polynomial power series expansions for entire axially monogenic functions. In relation to the difference and sum bases of monogenic polynomials, which have link with Bernoulli polynomials, the best possible bounds for the orders of such bases are determined. Moreover the Tρ-property of such differenced base and the sum base is discussed. The results obtained contain previous ones of complex setting as special cases. [ABSTRACT FROM AUTHOR]

In the first part of this article, we give the definition of bihypermonogenic functions in Clifford analysis. Using the idea of quasi-permutation, introduced by Sha Huang [Quasi-permutations and generalized regular functions in real Clifford analysis, J. Sys. Sci. and Math. Sci 18 (1998), pp. 380-384], we state an equivalent condition for bihypermonogenicity. In the second part, we discuss the Cauchy integral formula and Plemelj formula for the bihypermonogenic functions in real Clifford analysis. [ABSTRACT FROM AUTHOR]

Let [image omitted] be polynomials in [image omitted] and let [image omitted] be the real algebraic set associated with Q( [image omitted]) and let [image omitted] be a compact subset of the algebraic set Q. We describe an algebraic approach for solving the general boundary value problem (BVP): given partial differential equation (PDE) [image omitted] and a continuous function [image omitted] find [image omitted] so that [image omitted] We will show how the general technique applies in the case that P(y) is a homogeneous polynomial of degree deg(P(y)) and [image omitted], where [image omitted] is a polynomial having deg [image omitted] < deg (P(y)) and prove that the solution is unique in this case. This article brings together ideas from partial differential equations, a generalization of the theory of functions of a complex variable and the theory of commutative algebras. [ABSTRACT FROM AUTHOR]

Using three different representations of the bicomplex numbers [image omitted], which is a commutative ring with zero divisors defined by [image omitted] where i_1² = - 1, i_2² = - 1, j2 = 1 and i1i2 = j = i2i1, we construct three classes of bicomplex pseudoanalytic functions. In particular, we obtain some specific systems of Vekua equations of two complex variables and we established some connections between one of these systems and the classical Vekua equations. We also consider the complexification of the real stationary two-dimensional Schrodinger equation. With the aid of any of its particular solutions, we construct a specific bicomplex Vekua equation possessing the following special property. The scalar parts of its solutions are solutions of the original complexified Schrodinger equation and the vectorial parts are solutions of another complexified Schrodinger equation. [ABSTRACT FROM AUTHOR]

By using the higher-order Cauchy-Pompeiu formula for functions taking values in a Clifford algebra, the class of generalized regular functions is defined. Some properties, such as the Cauchy integral representation, the Cauchy theorem, Momera's theorem, total analyticity and extension theorem of Hartog's type from theory of regular functions in Clifford analysis, are generalized for those functions. [ABSTRACT FROM AUTHOR]

In [Gentili, G. and Stoppato, C., 2007, Zeros of regular functions and polynomials of a quaternionic variable, n. 1 (Dipartimento di Matematica "U. Dini", Università di Firenze), Preprint; Gentili, G. and Struppa, D.C., 2006, A new approach to Cullen-regular functions of a quaternionic variable. C. R. Acad. Sci. Paris, Ser. I, 342, 741-744; Gentili, G. and Struppa, D.C., 2007, A new theory of regular functions of a quaternionic variable. Adv. Math., 216, 279-301.] the authors offered an alternative definition and theory of regularity for functions of a quaternionic or octonionic variable, inspired by an idea of Cullen [Cullen, C.G., 1965, An integral theorem for analytic intrinsic functions on quaternions. Duke Mathematics Journal, 32, 139-148]. This alternative theory is intriguing because it allows the study of natural power series (and polynomials) with quaternionic or octonionic coefficients, which is excluded when the Fueter approach is followed. In this article the authors follow the same ideas to offer a new definition of regularity on the Clifford algebra Cl(0, 3). It turns out that this setting presents a family of new phenomena, which did not appear in the cases of quaternions and octonions. It is nevertheless possible to find a power series expansion for functions regular on an appropriate subset of Cl(0, 3). [ABSTRACT FROM AUTHOR]

Two kinds of integral properties in Clifford analysis are considered. The first deals with the boundary properties of the Cauchy singular integral when the boundary is composed of piecewise smooth surfaces. The second discusses the connections between the weighted integral operators and the orthogonal polynomials. Both are closely connected with the Borel-Pompeiu formula. [ABSTRACT FROM AUTHOR]

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 [image omitted] on the underlying vector space, eventually extended to the whole complex Clifford algebra [image omitted]. 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. [ABSTRACT FROM AUTHOR]

The theory of functions with values in Clifford algebras shows a lot of analogies to the complex function theory in the complex one-dimensional case. The class of holomorphic functions is now the set of null solutions of a generalized Cauchy-Riemann system, the class of monogenic functions. Analogously to the complex case one can define a derivative of monogenic functions by applying the adjoint Cauchy-Riemann operator. The main goal of this article is to study whether the derivative can be "inverted" and how monogenic primitives of monogenic functions can be constructed. [ABSTRACT FROM AUTHOR]

In this article, by defining the generalized co-dimension-p sinc function, the corresponding sinc interpolations (Shannon sampling theorems) are obtained. [ABSTRACT FROM AUTHOR]

We define and characterize rational hyperholomorphic functions from Rn+1 into the associated Clifford algebra. [ABSTRACT FROM AUTHOR]

Based on the solution of the Helmholtz equation in this article, we get the generalized Cauchy integral formula, the Cauchy–Pompeiu formula, the higher-order Cauchy–Pompeiu formula, and the generalized integral representations related with the Helmholtz operator for functions with values in a universal Clifford algebra C(V n ,n) . We also give weak solutions of the inhomogeneous equations L k u &ThickSpace;=&ThickSpace; f and , k &ThickSpace;≥&ThickSpace;1, where Lu=Du+uh and , D is the Dirac operator. [ABSTRACT FROM AUTHOR]

This article deals with the biregular functions on unbounded domains in Clifford analysis. We study its Plemelj formula and a linear boundary value problem By using the integral equation method and the compression fixed point theorem, the existence and the uniqueness of the solution for the problem are proved. [ABSTRACT FROM AUTHOR]

This paper is an updated version of our former article (Sommen, F., 1998, Curved Radon transforms in Clifford analysis. In: Clifford Algebras and their Applications in Mathematical Physics, Fund. Theories Phys., Vol. 94 (Dordrecht: Kluwer Academic Publishers), pp. 369–381). First of all, we study weighted integrals of functions over general surfaces of higher codimension whereby the weights take values in a Clifford algebra. We also introduce and study multi-linear Grassmann and Clifford algebras and apply them to the multi-linear Radon transform. In cases of symmetric multi-linear functions (homogeneous polynomials), we obtain a Clifford analysis generalization of the generalized Radon transform investigated by V.P. Palamodov (Palamodov, V.P., 1994, Radon transformation on real algebraic varieties. In: Gindikin, S. and Michor, P. (Eds) 75 Years of Radon Transform, Lecture Notes in Mathematical Physics, Vol. IV (Boston: International Press), pp. 252–262). [ABSTRACT FROM AUTHOR]

In this article we establish the basis for a Clifford analysis over finite reflection groups. [ABSTRACT FROM AUTHOR]

This article deals with the Hardy space related to solutions of generalized Rarita-Schwinger operators in the ball and the half space. These operators generalize the Rarita-Schwinger operator for spin 3⁄2-fields to the case of functions taking values in irreducible representation spaces with weight k + ½. [ABSTRACT FROM AUTHOR]

In this article, half Robin problems for powers of Dirac operator in the unit ball B of Rm (m x≥ 3) are investigated. By using a certain kind of decomposition of k-monogenic functions an explicit formula for the solution and the necessary and sufficient conditions for the solvability of the problems are obtained. [ABSTRACT FROM AUTHOR]

Let Cℓn be the (universal) Clifford algebra generated by el, ⋯ en, satisfying eiej + ejei = -2δij, i, j = 1, ⋯ n. The Dirac operator in Cℓn is defined by D = ∑i=0nei(∂/∂xi), where e0 = 1. The modified Dirac operator is introduced in (Eriksson-Bique and Leutwiler 2000, Hypermonogenic functions. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2 (Boston: Birkhäuser), pp. 287–302) by Mƒ = Dƒ + k(Q′ƒ/xn), where ′ denotes the main involution in Cℓn and Qƒ is given by the decomposition f(x) = Pƒ(x) + Qƒ(x)en with Pƒ(x), Qƒ(x) ∊ Cℓn-1. A continuously differentiable function f: Ω → Cℓn is called hypermonogenic in an open subset Ω of ℝn+1, if xn Mƒ(x) = 0, for all x ∊ Ω. Paravector-valued hyper-monogenic functions are called H-solutions, see (Leutwiler, 1992, Modified Clifford analysis, Complex Variables, 17, 153–171). The power function xm (m ∊ ℕ0) is an H-solution. We give a Cauchy-type formula for H-solutions. Furthermore we derive the equation D(Δ(n-2)/2) g) = 0 for the restriction g of the hypermonogenic function ƒ to the hyperplane xn = 0. This equation has been considered by Laville, Lehman and Ramadanoff, see (2004, Analytic Cliffordian functions, Annales Academiae Scientiarium Fennicae. Mathematica, 29(2), 251–268; 1998, Holomorphic Cliffordian functions, Advances in Applied Clifford Algebras, 8(2), 323–340) and in a different context by J. Ryan, see (1990, Iterated Dirac operators in Cn, Zeitschrift fur Analysis und ihre Anwendungen, 9(5), 385–401), and others. [ABSTRACT FROM AUTHOR]

Integral representation formulae are developed in Clifford analysis with respect to powers of the Dirac operator. Both C(Vm, 0) and C(Vm, m) are considered. The Helmholtz operator, its factors and their powers are likewise investigated. [ABSTRACT FROM AUTHOR]

Let Γ be an n-dimensional rectifiable ∣F∣-regular surface in ℝn+1. Let u be a continuous ℝ0, n- valued function on Γ, where ℝ0, n is the Clifford algebra associated with ℝn. Then we prove that the weighted Cliffordian Cauchy transform (Multiple line equation(s) cannot be represented in ASCII text) has continuous limit values on F if and only if the truncated integrals (Multiple line equation(s) cannot be represented in ASCII text) converge uniformly in Γ as ϵ → 0. [ABSTRACT FROM AUTHOR]