# American Institute of Mathematical Sciences

July  2011, 10(4): 1097-1109. doi: 10.3934/cpaa.2011.10.1097

## The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis

 1 Cli ord Research Group, Faculty of Engineering, Ghent University, Galglaan 2, 9000, Gent, Belgium 2 Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium 3 Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent

Received  October 2010 Revised  January 2011 Published  April 2011

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this paper, we establish a Cauchy-Kovalevskaya extension theorem for discrete monogenic functions defined on the standard $Z^m$ grid. Based on this extension principle, discrete Fueter polynomials, forming a basis of the space of discrete spherical monogenics, i.e. homogeneous discrete monogenic polynomials, are introduced. As an illustrative example we moreover explicitly construct the Cauchy-Kovalevskaya extension of the discrete delta function. These results are then generalized for a grid with variable mesh width $h$.
Citation: Hilde De Ridder, Hennie De Schepper, Frank Sommen. The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1097-1109. doi: 10.3934/cpaa.2011.10.1097
##### References:
 [1] F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Research Notes in Mathematics, 76, Pitman, London, 1982. [2] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: an overview, Cubo, 11 (2009), 55-71. [3] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: a germ of function theory, In: I. Sabadini, M. Shapiro, F. Sommen (eds.), Hypercomplex Analysis, Birkhäuser, (2009), 37-53. [4] R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions - A Function Theory for the Dirac Operator," Kluwer Academic Publishers, Dordrecht, 1992. [5] A. Cauchy, Oeuvres completes, Série 1, Tome VII, Gauthier-Villars, Paris, 1882-1974, 17-58. [6] R. Cooke, The Cauchy-Kovalevskaya Theorem (preprint, available online: http://www.cems.uvm.edu/ cooke/ckthm.pdf). [7] H. De Ridder, H. De Schepper, F. Sommen and U. Kähler, Discrete function theory based on skew Weyl relations, Proc. Amer. Math. Soc., 138 (2010), 3241-3256. [8] H. De Ridder, H. De Schepper and F. Sommen, Fueter polynomials in discrete Clifford analysis, (submitted). [9] N. Faustino, U. Kähler and F. Sommen, Discrete Dirac operators in Clifford analysis, Adv. Appl. Cliff. Alg., 17 (2007), 451-467. [10] J. Gilbert and M. Murray, "Clifford Algebra and Dirac Operators in Harmonic Analysis," Cambridge University Press, Cambridge, 1991. [11] K. Gürlebeck and W. Sprössig, "Quaternionic and Clifford Calculus for Physicists and Engineers," J. Wiley & Sons, Chichester, 1997. [12] S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung, J. für die Reine und Angew. Mathem., 80 (1875), 1-32.

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##### References:
 [1] F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis," Research Notes in Mathematics, 76, Pitman, London, 1982. [2] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: an overview, Cubo, 11 (2009), 55-71. [3] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: a germ of function theory, In: I. Sabadini, M. Shapiro, F. Sommen (eds.), Hypercomplex Analysis, Birkhäuser, (2009), 37-53. [4] R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions - A Function Theory for the Dirac Operator," Kluwer Academic Publishers, Dordrecht, 1992. [5] A. Cauchy, Oeuvres completes, Série 1, Tome VII, Gauthier-Villars, Paris, 1882-1974, 17-58. [6] R. Cooke, The Cauchy-Kovalevskaya Theorem (preprint, available online: http://www.cems.uvm.edu/ cooke/ckthm.pdf). [7] H. De Ridder, H. De Schepper, F. Sommen and U. Kähler, Discrete function theory based on skew Weyl relations, Proc. Amer. Math. Soc., 138 (2010), 3241-3256. [8] H. De Ridder, H. De Schepper and F. Sommen, Fueter polynomials in discrete Clifford analysis, (submitted). [9] N. Faustino, U. Kähler and F. Sommen, Discrete Dirac operators in Clifford analysis, Adv. Appl. Cliff. Alg., 17 (2007), 451-467. [10] J. Gilbert and M. Murray, "Clifford Algebra and Dirac Operators in Harmonic Analysis," Cambridge University Press, Cambridge, 1991. [11] K. Gürlebeck and W. Sprössig, "Quaternionic and Clifford Calculus for Physicists and Engineers," J. Wiley & Sons, Chichester, 1997. [12] S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung, J. für die Reine und Angew. Mathem., 80 (1875), 1-32.
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