77 research outputs found
Monogenic Gaussian distribution in closed form and the Gaussian fundamental solution
In this paper we present a closed formula for the CK-extension of the
Gaussian distribution in , and the monogenic version of the
holomorphic function which is a fundamental solution of the
generalized Cauchy-Riemann operator
Fueter's theorem: the saga continues
In this paper is extended the original theorem by Fueter-Sce (assigning an
-valued monogenic function to a -valued holomorphic
function) to the higher order case. We use this result to prove Fueter's
theorem with an extra monogenic factor .Comment: 11 pages, accepted for publication in Journal of Mathematical
Analysis and Application
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
Biaxial monogenic functions from Funk-Hecke's formula combined with Fueter's theorem
Funk-Hecke's formula allows a passage from plane waves to radially invariant
functions. It may be adapted to transform axial monogenics into biaxial
monogenics that are monogenic functions invariant under the product group
SO(p)xSO(q). Fueter's theorem transforms holomorphic functions in the plane
into axial monogenics, so that by combining both results, we obtain a method to
construct biaxial monogenics from holomorphic functions.Comment: 12 page
Orthogonality of Hermite polynomials in superspace and Mehler type formulae
In this paper, Hermite polynomials related to quantum systems with orthogonal
O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry
Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview
of the results for O(m) and G, the orthogonality of the Hermite polynomials
related to Sp(2n) is obtained with respect to the Berezin integral. As a
consequence, an extension of the Mehler formula for the classical Hermite
polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a
full superspace with O(m) x Sp(2n) symmetry are considered. It is shown that
they are not orthogonal with respect to the canonically defined inner product.
However, a new inner product is introduced which behaves correctly with respect
to the structure of harmonic polynomials on superspace. This inner product
allows to restore the orthogonality of the Hermite polynomials and also
restores the hermiticity of a class of Schroedinger operators in superspace.
Subsequently, a Mehler formula for the full superspace is obtained, thus
yielding an eigenfunction decomposition of the super Fourier transform.
Finally, an extensive comparison is made of the results in the different types
of symmetry.Comment: Proc. London Math. Soc. (2011) (42pp
Reproducing kernels for polynomial null-solutions of Dirac operators
It is well-known that the reproducing kernel of the space of spherical
harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going
over to complex variables and restricting to suitable bihomogeneous subspaces,
one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads
to Koornwinder's celebrated result on the addition formula.
In the present paper, the space of Hermitian monogenics, which is the space
of polynomial bihomogeneous null-solutions of a set of two complex conjugated
Dirac operators, is considered. The reproducing kernel for this space is
obtained and expressed in terms of sums of Jacobi polynomials. This is achieved
through use of the underlying Lie superalgebra , combined
with the equivalence between the inner product on the unit sphere and the
Fischer inner product. The latter also leads to a new proof in the standard
Dirac case related to the Lie superalgebra .Comment: 31 pages, 1 table, to appear in Constr. Appro
On two-sided monogenic functions of axial type
In this paper we study two-sided (left and right) axially symmetric solutions
of a generalized Cauchy-Riemann operator. We present three methods to obtain
special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave
integrals and Funk-Hecke's formula and via primitivation. Each of these methods
is effective enough to generate all the polynomial solutions.Comment: 17 pages, accepted for publication in Moscow Mathematical Journa
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