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Discrete Clifford Analysis: A Germ of Function Theory

  • F. Brackx
  • H. De Schepper
  • F. Sommen
  • L. Van de Voorde
Part of the Trends in Mathematics book series (TM)

Abstract

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes’ theorem, Cauchy’s theorem and a Cauchy integral formula are established.

Keywords

Discrete Clifford analysis discrete function theory discrete Cauchy integral formula 

Mathematics Subject Classification (2000)

Primary 30G35 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • F. Brackx
    • 1
  • H. De Schepper
    • 1
  • F. Sommen
    • 1
  • L. Van de Voorde
    • 1
  1. 1.Clifford Research Group Faculty of EngineeringGhent UniversityGentBelgium

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