On homogeneous polynomial solutions of generalized MoisilThéodoresco systems in Euclidean space
 Author
 Richard Delanghe (UGent)
 Organization
 Abstract
 Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , R(s)_0,m+1 be the space of svectors 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 MoisilTheodoresco 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+1valued 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, MoisilTheodoresco systems, Clifford analysis, conjugate harmonic funtions
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Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU1156857
 Chicago
 Delanghe, Richard. 2010. “On Homogeneous Polynomial Solutions of Generalized MoisilThéodoresco Systems in Euclidean Space.” Cubo  A Mathematical Journal 12 (2): 145–167.
 APA
 Delanghe, R. (2010). On homogeneous polynomial solutions of generalized MoisilThéodoresco systems in Euclidean space. CUBO  A MATHEMATICAL JOURNAL, 12(2), 145–167.
 Vancouver
 1.Delanghe R. On homogeneous polynomial solutions of generalized MoisilThéodoresco systems in Euclidean space. CUBO  A MATHEMATICAL JOURNAL. 2010;12(2):145–67.
 MLA
 Delanghe, Richard. “On Homogeneous Polynomial Solutions of Generalized MoisilThéodoresco 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 svectors 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 MoisilTheodoresco 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+1valued 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 = {07167776}, journal = {CUBO  A MATHEMATICAL JOURNAL}, keyword = {polynomial bases,harmonic potentials,MoisilTheodoresco systems,Clifford analysis,conjugate harmonic funtions}, language = {eng}, number = {2}, pages = {145167}, title = {On homogeneous polynomial solutions of generalized MoisilTh{\'e}odoresco systems in Euclidean space}, volume = {12}, year = {2010}, }