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

The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis

Hilde De Ridder 1, , Hennie De Schepper 2, and  Frank Sommen 3,

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 $h$.
Keywords: discrete Clifford analysis., Cauchy-Kovalevskaya extension.
Mathematics Subject Classification: Primary: 39A12, 30G3.
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.

show all references

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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