CUBO A Mathematical Journal Vol.12, N° 02, (189-197). June 2010

Fischer decomposition by inframonogenic functions

Helmuth R. Malonek¹, Dixan Peña Peña² and Frank Sommen

Department of Mathematics, Aveiro University, 3810-193 Aveiro, Portugal ¹email: hrmalon@ua.pt ² email: dixanpena@ua.pt, dixanpena@gmail.com

Department of Mathematical Analysis, Ghent University, 9000 Gent, Belgium email: fs@cage.ugent.be

ABSTRACT

Let δ_x denote the Dirac operator in R^m. In this paper, we present a refinement of the biharmonic functions and at the same time an extension of the monogenic functions by considering the equation δ_xf δ_x = 0. The solutions of this “sandwich” equation, which we call inframonogenic functions, are used to obtain a new Fischer decomposition for homogeneous polynomials in R^m.

Key words and phrases: Inframonogenic functions; Fischer decomposition.

RESUMEN

Denotemos por δ_x el operador de Dirac en R^m. En este artículo, nosotros presentamos un refinamiento de las funciones biarmónicas y al mismo tiempo una extensión de las funciones monogénicas considerando la ecuación δ_xfδ_x = 0. Las soluciones de esta ecuación tipo “sándwich”, las cuales llamaremos inframonogénicas, son utilizadas para obtener una nueva descomposición de Fischer para polinomios homogéneos en R^m.

Mathematics Subject Classification: 30G35; 31B30; 35G05.

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Received: March 2009.

Revised: May 2009.