481 research outputs found
Orthogonal Appell bases for Hodge-de Rham systems in Euclidean spaces
Recently the Gelfand-Tsetlin construction of orthogonal bases has been
explicitly described for the spaces of k-homogeneous polynomial solutions of
the Hodge-de Rham system in the Euclidean space R^m which take values in the
space of s-vectors. In this paper, we give another construction of these bases
and, mainly, we show that the bases even form complete orthogonal Appell
systems. Moreover, we study the corresponding Taylor series expansions. As an
application, we construct quite explicitly orthogonal bases for homogeneous
solutions of an arbitrary generalized Moisil-Theodoresco system.Comment: submitte
The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces
The main aim of this paper is to construct explicitly orthogonal bases for
the spaces of k-homogeneous polynomial solutions of the Hodge-de Rham system in
the Euclidean space R^m which take values in the space of s-vectors. Actually,
we describe even the so-called Gelfand-Tsetlin bases for such spaces in terms
of Gegenbauer polynomials. As an application, we obtain an algorithm how to
compute an orthogonal basis of the space of homogeneous solutions of a
generalized Moisil-Theodoresco system in R^m.Comment: submitte
On primitives and conjugate harmonic pairs in hermitian Clifford analysis
The notion of a conjugate harmonic pair in the context of Hermitian Clifford analysis is introduced as a pair of specific harmonic functions summing up to a Hermitian monogenic function in an open region of . Hermitian monogenic functions are special monogenic functions, which are at the core of so-called Clifford analyis, a straightforward generalization to higher dimension of the holomorphic functions in the complex plane. Under certain geometric conditions on the conjugate harmonic to a given specific harmonic is explicitly constructed and the potential or primitive of a Hermitian monogenic function is determined
Fundaments of Quaternionic Clifford Analysis II: Splitting of Equations
Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a
higher dimensional function theory which refines harmonic analysis and
generalizes to higher dimension the theory of holomorphic functions in the
complex plane. So-called quaternionic monogenic functions satisfy a system of
first order linear differential equations expressed in terms of four
interrelated Dirac operators. The conceptual significance of quaternionic
Clifford analysis is unraveled by showing that quaternionic monogenicity can be
characterized by means of generalized gradients in the sense of Stein and
Weiss. At the same time, connections between quaternionic monogenic functions
and other branches of Clifford analysis, viz Hermitian monogenic and standard
or Euclidean monogenic functions are established as well
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