# 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
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