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Las álgebras de Clifford son álgebras asociativas y no conmutativas definidas a través de ciertas estructuras multiplicativas. En estas álgebras no siempre existe una fórmula explícita que permita expresar el producto entre los vectores de la base del espacio vectorial, tal como está propuesto en el álgebra An (ver [6]). En esta investigación se ofrece una expresión explícita para el producto de determinados elementos de la base del álgebra An(2, αj , γij ), lo cual representa la apertura para deducir cálculos que arrojen nuevos aportes en el análisis de Clifford con parámetros.

In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $\mathbb{R}^m$. Being defined as the solutions of elliptic (generally non-strongly elliptic) partial differential equations, $(\varphi,\psi)$-inframonogenic and $(\varphi,\psi)$-harmonic functions do not share the good structure and properties of the harmonic ones. The aim of this paper it to show and clarified the relationship between these classes of functions., Comment: 11 pages, 2 figures, paper

As an integral representation for holomorphic functions, Cauchy integral formula plays a significant role in the function theory of one complex variable and several complex variables. In this paper, using the idea of several complex analysis we construct the Cauchy kernel in universal Clifford analysis, which has generalized complex differential forms with universal Clifford basic vectors. We establish Cauchy–Pompeiu formula and Cauchy integral formula on the distinguished boundary with values in universal Clifford algebra. This work is the basis for studying the Cauchy-type integral and its boundary value problem in complex universal Clifford analysis.

In this paper we consider generalized Hardy spaces in the octonionic setting associated to arbitrary Lipschitz domains where the unit normal field exists almost everywhere. First we discuss some basic properties and explain structural differences to the associative Clifford analysis setting. The non-associativity requires special attention in the definition of an appropriate inner product and hence in the definition of a generalized Szegö projection. Whenever we want to apply classical theorems from reproducing kernel Hilbert spaces we first need to switch to the consideration of real-valued inner products where the Riesz representation theorem holds. Then we introduce a generalization of the dual Cauchy transform for octonionic monogenic functions which represents the adjoint transform with respect to the real-valued inner product $$\langle \cdot , \cdot \rangle _0$$ ⟨ · , · ⟩ 0 together with an associated octonionic Kerzman–Stein operator and related kernel functions. Also in the octonionic setting, the Kerzman–Stein operator that we introduce turns out to be a compact operator. A motivation behind this approach is to find an approximative method to compute the Szegö projection of octonionic monogenic functions offering a possibility to tackle BVP in the octonions without the explicit knowledge of the octonionic Szegö kernel which is extremely difficult to determine in general. We also discuss the particular cases of the octonionic unit ball and the half-space. Finally, we relate our octonionic Kerzman–Stein operator to the Hilbert transform and particularly to the Hilbert–Riesz transform in the half-space case.

This paper aims at proving the boundedness property of multidimensional singular integral operators associated with $$(\varphi ,\psi )$$ -harmonic functions, which are connected by the use of two orthogonal basis of the Euclidean space $${{\mathbb {R}}}^m$$ . Besides, necessary and sufficient conditions for the solvability of the $${\overline{\partial }}$$ -problem for such $$(\varphi ,\psi )$$ -harmonic functions are described. The basic devices that make it possible to state and prove both results are borrowed from Clifford analysis.

The k-Cauchy–Fueter operator is an Euclidean version of the helicity k/2 massless field equations on affine Minkowski space. In this article, a version of Schwarz lemma associated to the k-Cauchy–Fueter is established by applying Bochner–Martinelli formula. The constant in Schwarz lemma is sharper than the former results when $$n = 1,k = 1$$ , the methods in Schwarz lemma may not only apply to the explicit dimension but also be valid for the general case which will generalize the results in Clifford analysis.

We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysis. In the physically significant case n=3, two explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lamé–Navier equation in bounded and unbounded domains are discussed.

Several topological methods have been used successfully in the study of the hypercomplex analysis; for example, in the theory of functions of several complex variables (Grauert et al., in: Encyclopaedia of mathematical science, vol. 74, Springer, 1991, Hirzebruch, in: Topological methods in algebraic geometry, Classics in Mathematics. Reprint of the 1978 Edition. Springer, Berlin, 1995, Krantz, in Function theory of several complex variables, 2nd edn. American Mathematical Society, Providence, 2001), in the Clifford analysis (Sabadini et al. in Adv. Appl. Clifford Algebras 24:1131–1143, 2014), and in the theory of slice regular functions (Colombo et al. in Math Nachr 285:949–958, 2012). Particularly, the fiber bundle is one of these topological subjects that had an intensive development in a number of papers (Bernstein and Philips in Sci Am 245(1):122–137, 1981, Bleecker, in: Guage Theory and Variational Principles. Dover Books on physics Dover Books on mathematics. Courier Corporation, North Chelmsford, 2005, Bredon in Topology and geometry, Springer, Berlin, 1913, Cohen in The topology of fiber bundles, Stanford University, Stanford, 1998, Hatcher in Algebraic-Topology, Cambridge University Press, Cambridge, 2002, Husemoller in fibre bundles, 3rd edn, Springer, Berlin, 1993, Steenrod in The topology of fibre bundles, Princeton University Press, Princeton, 1951, Walschap in Metric structures in differential geometry, Springer, New York, 2004, Weatherall in Synthese 193:2389–2425, 2016). The aim of this work is to show how the Splitting Lemma and the Representation Formula intrinsically determine a fiber bundle over the space of quaternionic slice regular functions and as a consequence, several properties of this function space are interpreted in terms of sections, pullbacks and isomorphism of fiber bundles.

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)-module. The main aim of the paper is to explain that a good possibility is to take the so-called generalized Cauchy-Riemann equations proposed a long time ago by E. Stein and G. Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)-modules. This is an important step in developing the corrresponding function theory.

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.

Nowadays, 3D images processing constitutes a challenging topic in many scientific fields such as medicine, computational physics, informatics. Therefore, the construction of suitable functional bases that allow computational aspects to be easily done is a necessity. Wavelets and Clifford algebras are ones of the most import mathematical tools for achieving such necessities. In the present work, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of new monogenic polynomials are provided based on two-parameter weight functions. Such classes englobe the well-known Gegenbauer and Hermite ones. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula and Fourier–Plancherel rules have been proved. Computational concrete examples are developed by means of some illustrative examples with graphical representations of the Clifford mother wavelets in some cases. These discovered wavelets have been applied by the next for modeling some biomedical signals such as EEG, ECG and proteins.

Using the method of normalized systems of functions, we study one representation of real analytic functions by monogenic functions (i.e., solutions of Dirac equations), which is an Almansi’s formula of infinite order. As applications of the representation, we construct solutions of the inhomogeneous Dirac and poly-Dirac equations in Clifford analysis.

In this paper we first define hypermonogenic solutions of the Dirac operator in R p × R q 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.