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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.

In this manuscript, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels., 38 pages

The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the biaxial Dirac operator $$\partial _{\mathbf{x}} +\partial _{\mathbf{y}}$$ . In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.

In this paper, we study a refinement of the Szego-Radon transform in the hypermonogenic setting. Hypermonogenic functions form a subclass of monogenic functions arising in the study of a modified Dirac operator, which allows for weaker symmetries and also has a strong connection to the hyperbolic metric. In particular, we construct a projection operator from a module of hypermonogenic functions in R p + q onto a suitable submodule of plane waves parameterized by a vector on the unit sphere of R q . Moreover, we study the interaction of this Szego-Radon transform with the generalized Cauchy-Kovalevskaya extension operator. Finally, we develop a reconstruction (inversion) method for this transform.

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.

In this paper, we study Pizzetti-type formulas for Stiefel manifolds and Cauchy-type formulas for the tangential Dirac operator from a distributional perspective. First we illustrate a general distributional method for integration over manifolds in $\mathbb R^m$ defined by means of $k$ equations $\varphi_1(\underline{x})=\ldots=\varphi_k(\underline{x})=0$. Next, we apply this method to derive an alternative proof of the Pizzetti formulae for the real Stiefel manifolds $SO(m)/SO(m-k)$. Besides, a distributional interpretation to invariant oriented integration is provided. In particular, we obtain a distributional Cauchy theorem for the tangential Dirac operator on an embedded $(m-k)$-dimensional smooth surface., Comment: 22 pages, small changes, updated reference list

There are two well-known ways of describing elements of the rotation group SO$(m)$. First, according to the Cartan-Dieudonn\'e theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix. In this paper, we study similar descriptions of a group of rotations SO${}_0$ in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp$(m|2n)$ on a Grassmann algebra. While still being connected, the group SO${}_0$ is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices. At the same time, SO${}_0$ strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO${}_0$. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO${}_0$. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group., Comment: 28 pages

We recall the algebra of endomorphisms on the so-called radial algebra of abstract vector variables which generalizes both polynomial and Clifford algebras, and the defining building blocks of a function theory in this abstract framework. These building blocks are given by the abstract versions of the Dirac operator and the directional derivatives. Together with other fundamental endomorphisms, they lead to an algebraic structure which can be considered as the abstract equivalent of orthogonal Clifford analysis. Following a similar approach, we present the axiomatic definitions of the Hermitian radial algebra, leading to the abstract equivalent of Hermitian Clifford analysis.

In this paper, we investigate the existence of a mixed Fischer decomposition in a generalized Clifford analysis setting, where a system of Dirac equations is considered, associated to different orthogonal bases (or structural sets) in Euclidean space.

The aim of this paper is to introduce, in the framework of Clifford analysis, the notions of \(\varphi \)-hyperdifferentiability and \(\varphi \)-hyperderivability for \(\psi \)-hyperholomorphic functions where (\(\varphi ,\psi \)) are two arbitrary orthogonal bases (called structural sets) of a Euclidean space. In this study we will also show how to exchange the integral sign and the \(\varphi \)-hyperderivative of the \(\psi \)-Cliffordian Cauchy-type integral. Thereby, we generalize, in a natural way, the corresponding quaternionic antecedent as well as the standard Clifford predecessor.

In this paper, we show that a higher order Borel–Pompeiu (Cauchy–Pompeiu) formula, associated with an arbitrary orthogonal basis (called structural set) of a Euclidean space, can be extended to the framework of generalized Clifford analysis. Furthermore, in lower dimensional cases, as well as for combinations of standard structural sets, explicit expressions of the kernel functions are derived.

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field . It relies heavily on results on functions defined on domains in or with values in . This theory is centred around the concept of ψ-hyperholomorphic functions related to a so-called structural set ψ of or respectively. The main goal of this paper is to develop the nucleus of the -hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy-Riemann operators associated to a pair of structural sets of . Following a matrix approach, a generalized Borel-Pompeiu formula and the corresponding Plemelj-Sokhotzki formulae are established.

Distributions in superspace constitute a very useful tool for establishing an integration theory. In particular, distributions have been used to obtain a suitable extension of the Cauchy formula to superspace and to define integration over the superball and the supersphere through the Heaviside and Dirac distributions, respectively. In this paper, we extend the distributional approach to integration over more general domains and surfaces in superspace. The notions of domain and surface in superspace are defined by smooth bosonic phase functions $g$. This allows to define domain integrals and oriented (as well as non-oriented) surface integrals in terms of the Heaviside and Dirac distributions of the superfunction $g$. It will be shown that the presented definition for the integrals does not depend on the choice of the phase function $g$ defining the corresponding domain or surface. In addition, some examples of integration over a super-paraboloid and a super-hyperboloid will be presented. Finally, a new distributional Cauchy-Pompeiu formula will be obtained, which generalizes and unifies the previously known approaches., Comment: 25 pages

In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO 0 leaving invariant such an inner product turns out to be an extension of SO ( m ) × Sp ( 2 n ) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO 0 by means of the representation h ( s ) [ x ] = s x s ‾ . In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3] . The group of complex supermatrices leaving this inner product invariant constitutes an extension of U ( m ) × U ( n ) and is isomorphic to the subset SO 0 J of SO 0 of elements that commute with the complex structure J. The realization of SO 0 J within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.

In this paper we first recall the proper algebraic framework, i.e. the radial algebra, needed to extend Hermitian Clifford analysis to the superspace setting. The fundamental objects for this extension then are introduced by means of an abstract complex structure on the Hermitian radial algebra. This leads to a natural representation of this Hermitian radial algebra on superspace.

In this paper we discuss the local solvability of the inhomogeneous Beltrami equations in Quaternionic Analysis. We give an example of a Beltrami equation with no distributional solution and deduce the compatibility condition. This study is closely linked to the study of Dirac operators with non-constant coefficients.

In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalize most of those previous results in this direction and additionally other consequent results are presented. In particular, the expression of the jump of the generalized Π-operator across the boundary of the domain is obtained as well as an estimate for the norm of the Π-operator is given. At the end an application of the generalized Π-operator to the solution of Beltrami equations is studied where we give conditions for a solution to realize a local and global homeomorphism.

The present paper is aimed at proving necessary and sufficient conditions on the quaternionic-valued coefficients of a first-order linear operator to be associated to the generalized Cauchy—Riemann operator in quarternionic analysis and explicitly we give the description of all its nontrivial first-order symmetries.