72 research outputs found
Distributions and Integration in superspace
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 . 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 . It will be shown that the presented definition for the
integrals does not depend on the choice of the phase function 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 page
Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace
The aim of this paper is to obtain a generalized CK-extension theorem in
superspace for the bi-axial Dirac operator. 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.Comment: 27 page
Pizzetti and Cauchy formulae for higher dimensional surfaces : a distributional approach
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 R^m defined by means of k equations phi(1)(x) = ... = phi(k)(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 of 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
Higher order Borel-Pompeiu representations
We establish a higher order Borel-Pompeiu formula for monogenic functions associated to an arbitrary orthogonal basis (called structural set) of Euclidean space, or combinations of such structural sets
Higher order Borel–Pompeiu representations in Clifford analysis
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
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
Spin actions in Euclidean and Hermitian Clifford analysis in superspace
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) x Sp(2n) 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] = sx\bar{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) X 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
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