# 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 & 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 (1982). [2] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: an overview,, Cubo, 11 (2009), 55. [3] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: a germ of function theory,, In: I. Sabadini, (2009), 37. [4] R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions - A Function Theory for the Dirac Operator,", Kluwer Academic Publishers, (1992). [5] A. Cauchy, Oeuvres completes,, S\'erie 1, (): 1882. [6] R. Cooke, The Cauchy-Kovalevskaya Theorem, (preprint, (). [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. [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. [10] J. Gilbert and M. Murray, "Clifford Algebra and Dirac Operators in Harmonic Analysis,", Cambridge University Press, (1991). [11] K. Gürlebeck and W. Sprössig, "Quaternionic and Clifford Calculus for Physicists and Engineers,", J. Wiley & Sons, (1997). [12] S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung,, J. f\, 80 (1875), 1.

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##### References:
 [1] F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis,", Research Notes in Mathematics, 76 (1982). [2] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: an overview,, Cubo, 11 (2009), 55. [3] F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: a germ of function theory,, In: I. Sabadini, (2009), 37. [4] R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions - A Function Theory for the Dirac Operator,", Kluwer Academic Publishers, (1992). [5] A. Cauchy, Oeuvres completes,, S\'erie 1, (): 1882. [6] R. Cooke, The Cauchy-Kovalevskaya Theorem, (preprint, (). [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. [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. [10] J. Gilbert and M. Murray, "Clifford Algebra and Dirac Operators in Harmonic Analysis,", Cambridge University Press, (1991). [11] K. Gürlebeck and W. Sprössig, "Quaternionic and Clifford Calculus for Physicists and Engineers,", J. Wiley & Sons, (1997). [12] S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung,, J. f\, 80 (1875), 1.
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