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On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space

(2010) CUBO - A MATHEMATICAL JOURNAL. 12(2). p.145-167
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Abstract
Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , R(s)_0,m+1 be the space of s-vectors in the Clifford algebra R_0,m+1 constructed over the quadratic vector space R^0,m+1 and let r, p, q, ∈ N be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of first order partial differential equations derived from the equation ∂W = 0 where W is R^(r,p,q)_0,m+1 valued and ∂ is the Dirac operator in R^m+1, is called a generalized Moisil-Theodoresco system of type (r, p, q) in Rm+1. For k ∈ N, k ≥ 1,MT+(m+1; k; R(r,p,q)_0,m+1), denotes the space of R(r,p,q)_0,m+1-valued homogeneous polynomials W_k of degree k in R^m+1 satisfying ∂W_k = 0. A characterization of W_k ∈ MT is given in terms of a harmonic potential H_k+1 belonging to a subclass of R(r,p,q)_0,m -valued solid harmonics of degree (k + 1) in Rm+1. Furthermore, it is proved that each W_k ∈MT admits a primitive W_k+1. Special attention is paid to the lower dimensional cases R3 and R4. In particular, a method is developed for constructing bases for the spaces MT, for even r.
Keywords
polynomial bases, harmonic potentials, Moisil-Theodoresco systems, Clifford analysis, conjugate harmonic funtions

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Chicago
Delanghe, Richard. 2010. “On Homogeneous Polynomial Solutions of Generalized Moisil-Théodoresco Systems in Euclidean Space.” Cubo - A Mathematical Journal 12 (2): 145–167.
APA
Delanghe, R. (2010). On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space. CUBO - A MATHEMATICAL JOURNAL, 12(2), 145–167.
Vancouver
1.
Delanghe R. On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space. CUBO - A MATHEMATICAL JOURNAL. 2010;12(2):145–67.
MLA
Delanghe, Richard. “On Homogeneous Polynomial Solutions of Generalized Moisil-Théodoresco Systems in Euclidean Space.” CUBO - A MATHEMATICAL JOURNAL 12.2 (2010): 145–167. Print.
@article{1156857,
  abstract     = {Let for s \ensuremath{\in} \{0, 1, ...,m+ 1\} (m \ensuremath{\geq} 2) , R(s)\_0,m+1 be the space of s-vectors in the Clifford algebra R\_0,m+1 constructed over the quadratic vector space R\^{ }0,m+1 and let r, p, q, \ensuremath{\in} N be
such that 0 \ensuremath{\leq} r \ensuremath{\leq} m + 1, p {\textlangle} q and r + 2q \ensuremath{\leq} m + 1. The associated linear system of first order partial differential equations derived from the equation \ensuremath{\partial}W = 0 where W is
R\^{ }(r,p,q)\_0,m+1 valued and \ensuremath{\partial} is the Dirac operator in R\^{ }m+1, is called a generalized Moisil-Theodoresco system of type (r, p, q) in Rm+1. For k \ensuremath{\in} N, k \ensuremath{\geq} 1,MT+(m+1; k; R(r,p,q)\_0,m+1), denotes the space of R(r,p,q)\_0,m+1-valued homogeneous polynomials W\_k of degree k in R\^{ }m+1 satisfying \ensuremath{\partial}W\_k = 0. A characterization of W\_k \ensuremath{\in} MT is given in terms of a harmonic potential H\_k+1 belonging to a subclass of R(r,p,q)\_0,m -valued
solid harmonics of degree (k + 1) in Rm+1. Furthermore, it is proved that each W\_k \ensuremath{\in}MT admits a primitive W\_k+1. Special
attention is paid to the lower dimensional cases R3 and R4. In particular, a method is developed for constructing bases for the spaces MT, for even r.},
  author       = {Delanghe, Richard},
  issn         = {0716-7776},
  journal      = {CUBO - A MATHEMATICAL JOURNAL},
  keyword      = {polynomial bases,harmonic potentials,Moisil-Theodoresco systems,Clifford analysis,conjugate harmonic funtions},
  language     = {eng},
  number       = {2},
  pages        = {145--167},
  title        = {On homogeneous polynomial solutions of generalized Moisil-Th{\'e}odoresco systems in Euclidean space},
  volume       = {12},
  year         = {2010},
}