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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000200012 

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 δx denote the Dirac operator in Rm. 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 δxf δx = 0. The solutions of this “sandwich” equation, which we call inframonogenic functions, are used to obtain a new Fischer decomposition for homogeneous polynomials in Rm.

Key words and phrases: Inframonogenic functions; Fischer decomposition.


RESUMEN

Denotemos por δx el operador de Dirac en Rm. En este artículo, nosotros presentamos un refinamiento de las funciones biarmónicas y al mismo tiempo una extensión de las funciones monogénicas considerando la ecuación δxx = 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 Rm.

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


References

[1] S. Bock and K. Gürlebeck, On a spatial generalization of the Kolosov-Muskhelishvili formulae, Math. Methods Appl. Sci. 32 (2009), no. 2, 223-240.        [ Links ]

[2] F. Brackx, On (k)-monogenic functions of a quaternion variable, Funct. theor. Methods Differ. Equat. 22-44, Res. Notes in Math., no. 8, Pitman, London, 1976.        [ Links ]

[3] F. Brackx, Non-(k)-monogenic points of functions of a quaternion variable, Funct. theor. Meth. part. Differ. Equat., Proc. int. Symp., Darmstadt 1976, Lect. Notes Math. 561, 138-149.        [ Links ]

[4] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis, Research Notes in Mathematics, 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.        [ Links ]

[5] P. Cerejeiras, F. Sommen and N. Vieira, Fischer decomposition and special solutions for the parabolic Dirac operator, Math. Methods Appl. Sci. 30 (2007), no. 9, 1057-1069.        [ Links ]

[6] W. K. Clifford, Applications of Grassmann's Extensive Algebra, Amer. J. Math. 1 (1878), no. 4, 350-358.        [ Links ]

[7] H. De Bie and F. Sommen, Fischer decompositions in superspace, Function spaces in complex and Clifford analysis, 170-188, Natl. Univ. Publ. Hanoi, Hanoi, 2008.        [ Links ]

[8] R. Delanghe, F. Sommen and V. Soucek, Clifford algebra and spinor-valued functions, Mathematics and its Applications, 53, Kluwer Academic Publishers Group, Dordrecht, 1992.        [ Links ]

[9] D. Eelbode, Stirling numbers and spin-Euler polynomials, Experiment. Math. 16 (2007), no. 1, 55-66.        [ Links ]

[10] N. Faustino and U. Kähler, Fischer decomposition for difference Dirac operators, Adv. Appl. Clifford Algebr. 17 (2007), no. 1, 37-58.        [ Links ]

[11] K. Gürlebeck and U. Kähler, On a boundary value problem of the biharmonic equation, Math. Methods Appl. Sci. 20 (1997), no. 10, 867-883.        [ Links ]

[12] H. R. Malonek and G. Ren, Almansi-type theorems in Clifford analysis, Math. Methods Appl. Sci. 25 (2002), no. 16-18, 1541-1552.        [ Links ]

[13] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem, Appl. Mech. Rev. 56 (2003), no. 1, 33-85.        [ Links ]

[14] G. Ren and H. R. Malonek, Almansi decomposition for Dunkl-Helmholtz operators, Wavelet analysis and applications, 35-42, Appl. Numer. Harmon. Anal., Birkhäuser, Basel, 2007.        [ Links ]

[15] J. Ryan, Basic Clifford analysis, Cubo Mat. Educ. 2 (2000), 226-256.        [ Links ]

[16] L. Sobrero, Theorie der ebenen Elastizität unter Benutzung eines Systems hyperkomplexerZahlen, Hamburg. Math. Einzelschriften, Leipzig, 1934.        [ Links ]

[17] F. Sommen, Monogenic functions of higher spin, Z. Anal. Anwendungen 15 (1996), no. 2, 279-282.        [ Links ]

[18] F. Sommen and N. Van Acker, Functions of two vector variables, Adv. Appl. Clifford Algebr. 4 (1994), no. 1, 65-72.        [ Links ]

Received: March 2009.

Revised: May 2009.