Dixan Peña

An alternative proof of Fueter's theorem

In this article we establish an alternative proof of the generalized Fueter method presented in a former paper [Qian, T. and Sommen, F., 2003, Deriving harmonic functions in higher dimensional spaces. Zeitschrift fur Analysis und ihre Anwendungen, 22(2), 275–288] leading to the construction of special harmonic and monogenic functions in higher dimensions. At the same time, we also obtain a generalization of this result. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Monogenic Gaussian distribution in closed form and the Gaussian fundamental solution

In this article we present a closed formula for the CK-extension of the Gaussian distribution in ℝ m , and the monogenic version of the holomorphic function which is a fundamental solution of the generalized Cauchy–Riemann operator.

Hermitean Cauchy Integral Decomposition of Continuous Functions on Hypersurfaces

We consider Hölder continuous circulant matrix functions defined on the Ahlfors-David regular boundary of a domain in . The main goal is to study under which conditions such a function can be decomposed as , where the components are extendable to two-sided -monogenic functions in the interior and the exterior of , respectively. -monogenicity is a concept from the framework of Hermitean Clifford analysis, a higher dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. -monogenic functions then are the null solutions of a matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context.

A Boundary Value Problem for Hermitian Monogenic Functions

We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in . Necessary and sufficient conditions for the solvability of this problem are obtained.

Fueter's theorem: The saga continues

Abstract In this paper is extended the original theorem by Fueter–Sce (assigning an R 0 , m -valued monogenic function to a C -valued holomorphic function) to the higher order case. We use this result to prove Fueters theorem with an extra monogenic factor P k ( x 0 , x ) .

On the jump problem for β-analytic functions

We develop the Plemelj–Sokhotski formulas for solutions of a special case of the Beltrami equation in the classical complex analysis. These formulas relate to a principal boundary value problem for β-analytic functions, the so-called jump problem. In connection with removable singularities for the β-analytic functions, the uniqueness of the solution of such a jump problem is examined. ¶Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.x

Some Power Series Expansions for Monogenic Functions

New series developments for monogenic functions are presented. The terms of these series have factors that are expressible as power functions vanishing on special higher codimension submanifolds of Euclidean space. These series are closely related with the Cauchy-Kowalewski extension problem as well as to special Vekua systems arising from the consideration of axial and biaxial symmetry.

RIEMANN BOUNDARY VALUE PROBLEM FOR HYPERANALYTIC FUNCTIONS

We deal with Riemann boundary value problem for hyperanalytic functions. Furthermore, necessary and sufficient conditions for solvability of the problem are derived. At the end the explicit form of general solution for singular integral equations with a hypercomplex Cauchy kernel in the Douglis sense is established.

Segal-Bargmann-Fock modules of monogenic functions

In this paper we introduce the classical Segal-Bargmann transform starting from the basis of Hermite polynomials and extend 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.

Full length article: Generating functions of orthogonal polynomials in higher dimensions

In this paper two important classes of orthogonal polynomials in higher dimensions using the framework of Clifford analysis are considered, namely the Clifford-Hermite and the Clifford-Gegenbauer polynomials. For both classes an explicit generating function is obtained.