Quaternionic k-Hyperbolic Derivative

Complex holomorphic functions are defined using a complex derivative. In higher dimensions the meaningful generalization of complex derivative is not straight forward. Sudbery defined a derivative for quaternion regular functions using differential forms. Gurlebeck and Malonek generalized that for monogenic functions. In this paper we find similar characterizations for k-hypermonogenic functions which are holomorphic functions based on the Riemannian metric ds(2) = dx(0)(2) + dx(1)(2) + dx(2)(2)/x(2)(2k) When k = 0 , we obtain the hypercomplex derivative by Gurlebeck and Malonek. Just like in the complex case derivative of k-hypermonogenic is the usual partial derivative with respect to the first coordinate.Peer reviewe

Hypermonogenic plane wave solutions of the dirac equation in superspace

In this paper, we obtain Cauchy-Kovalevskaya theorems for hypermonogenic superfunctions depending only on purely bosonic and fermionic vector variables. In addition, we use these results to construct plane wave examples of such functions

Vekua systems in hyperbolic harmonic analysis

In this paper we consider the solutions of the equation , where is the so called modifier Dirac operator acting on functions defined in the upper half-space and taking values in the Clifford algebra. We look for solutions where the first variable is invariant under rotations. A special type of solution is generated by the so called spherical monogenic functions. These solutions may be characterize by a Vekua-type system and this system may be solved using Bessel functions. We will see that the solution of the equation in this case will be a product of Bessel functions

Hyperbolic Function Theory in the Skew-Field of Quaternions

We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called alpha-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric ds(alpha)(2) = dx(0)(2) + dx(1)(2) + dx(2)(2) + dx(3)(2)/x(3)(alpha) in the upper half space R-+(4) = {( x(0), x(1), x(2), x(3)) is an element of R-4 : x(3) > 0}. If alpha = 2, the metric is the hyperbolic metric of the Poincare upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function x(m) (m is an element of Z), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental alpha-hyperbolic harmonic functions, depending only on the hyperbolic distance and x(3), we verify a Cauchy type integral formula for conjugate gradient of alpha-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued alpha-hypermonogenic in the Clifford algebra Cl-0,Cl-3.Peer reviewe

Two-sided hypergenic functions

In this paper we present an analogous of the class of two-sided axial monogenic functions to the case of axial $\kappa-$hypermonogenic functions. In order to do that we will solve a Vekua-type system in terms of Bessel functions

Structural results for quaternionic Gabor frames

We study quaternionic Gabor frames based on the two-sided quaternionic windowed Fourier transform. Since classical Hilbert space based methods do not work in this case we introduce appropriated versions of translation and modulation operators. We prove Janssen’s and Walnut’s representations, as well as modified versions of the Wexler–Raz biorthogonality and Ron–Shen duality based on the concept of correlation function. We end up with a characterization of tight quaternionic Gabor frames.publishe

New perspectives in hyperbolic function theory

In this thesis we are working with a function theory on the hyperbolic upper-half space. The function theory is called the hyperbolic function theory and it is studied since 1990's by Heinz Leutwiler and Sirkka-Liisa Eriksson. The advantage of the hyperbolic function theory is that positive and negative powers of hypercomplex variables are included to the theory. Thus the hyperbolic function theory offers a natural generalization of classical complex analysis

Symmetries in quaternionic analysis

This survey-type paper deals with the symmetries related to quaternionic analysis. The main goal is to formulate an SU(2) invariant version of the theory. First, we consider the classical Lie groups related to the algebra of quaternions. After that, we recall the classical Spin(4) invariant case, that is Cauchy–Riemann operators, and recall their basic properties. We define the SU(2) invariant operators called the Coifman– Weiss operators. Then we study their relations with the classical Cauchy–Riemann operators and consider the factorization of the Laplace operator. Using SU(2) invariant harmonic polynomials, we obtain the Fourier series representations for quaternionic valued functions studying in detail the matrix coefficients.publishedVersionPeer reviewe

Some Theoretical Remarks of Octonionic Analysis

In this article we first review the classical results of octonions and octonionic analysis. Then we consider some theoretical properties of the theory and compare it to quaternionic analysis and Clifford analysis.publishedVersionPeer reviewe

Cauchy–Riemann Operators in Octonionic Analysis

In this paper we first recall the definition of the octonion algebra and its algebraic properties. We derive the so called e4-calculus and using it we obtain the list of generalized Cauchy–Riemann systems for octonionic monogenic functions.acceptedVersionPeer reviewe